Ordinary Differential Equations

Transcription

1 58229_CH0_00_03.indd Page 6/6/6 2:48 PM F-007 /202/JB0027/work/indd & Bartlett Learning LLC, an Ascend Learning Compan.. PART Ordinar Differential Equations. Introduction to Differential Equations 2. First-Order Differential Equations 3. Higher-Order Differential Equations 4. The Laplace Transform 5. Series Solutions of Linear Differential Equations of Ordinar Differential Equations 6. Numerical Solutions & Bartlett Learning, LLC Nessa Gnatoush/Shutterstock

2 58229_CH0_00_03.indd Page 2 6/6/6 2:48 PM F-007 /202/JB0027/work/indd & Bartlett Learning LLC, an Ascend Learning Compan.. And Zarivn/ShutterStock, Inc.

3 & Bartlett Learning LLC, an Ascend Learning Compan.. CHAPTER Introduction to Differential Equations The purpose of this short chapter & is twofold: Bartlett to Learning, introduce the basic LLC CHAPTER CONTENTS Jones & Bartlett Learning, LLC NOT FOR terminolog SALE OR of differential DISTRIBUTION equations and to briefl eamine how differential equations arise. Definitions and Terminolog in an attempt to describe or.2 Initial-Value Problems model phsical phenomena in.3 Differential Equations as Mathematical Models mathematical terms. Chapter in Review NOT FOR SALE OR DISTRIBUTI

4 & Bartlett Learning LLC, an Ascend Learning Compan.. 4 CHAPTER Introduction to Differential Equations. Definitions and Terminolog INTRODUCTION The words differential and equation certainl suggest solving some kind of equation that contains derivatives. But before ou start solving anthing, ou must learn some of the basic defintions and terminolog of the subject. A & Definition Bartlett Learning, The derivative LLC d/d of a function f() is Jones itself another & Bartlett function f () Learning, LLC NOT FOR found SALE b an appropriate OR DISTRIBUTION rule. For eample, the function e NOT 0.2 is differentiable FOR SALE on the OR interval DISTRIBUTIO ( q, q), and its derivative is d/d 0.2e 0.2. If we replace e 0.2 in the last equation b the smbol, we obtain d 0.2. () d Now imagine that a friend of ours simpl hands ou the differential equation in (), and that ou have no idea how it was constructed. Your friend asks: What is the function represented b the smbol? You are now face-to-face with one of the basic problems in a course in differential equations: How do ou solve such an equation Jones for & the Bartlett unknown Learning, function f()? LLC The problem is loosel equivalent to the familiar reverse problem of differential calculus: Given a derivative, find an antiderivative. Before proceeding an further, let us give a more precise definition of the concept of a differential equation. NOT FOR Definition SALE.. OR DISTRIBUTION Differential Equation An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation (DE). In order to talk about them, we will classif a NOT differential FOR equation SALE b OR tpe, DISTRIBUTION order, and linearit. Classification b Tpe If a differential equation contains onl ordinar derivatives of one or more functions with respect to a single independent variable it is said to be an ordinar differential equation (ODE). An equation involving onl partial derivatives of one or more functions of two or more independent variables & Bartlett is called Learning, a partial differential LLC equation (PDE). Our first eample illustrates NOT several FOR of each SALE tpe of OR differential DISTRIBUTION equation. EXAMPLE (a) The equations d d 6 e, Tpes of Differential Equations an ODE can Jones contain more & Bartlett Learning, LLC than one dependent variable NOT T FOR T SALE OR DISTRIBUTI d 2 d d d 2 2 0, and 3 2 (2) 2 d d dt dt are eamples of ordinar differential equations. (b) The equations 0 2 u 0 02 u 0 2 u 0, u 2 0t 2 0u 2 0t, 0u 0 0v (3) 0 are eamples of partial differential equations. Notice in the third equation that there are two dependent variables and two Jones independent & Bartlett variables Learning, the PDE. This LLCindicates that u and v must be functions of two NOT or more FOR independent SALE variables. OR DISTRIBUTION

5 & Bartlett Learning LLC, an Ascend Learning Compan.. Notation Throughout this tet, ordinar derivatives will be written using either the Leibniz notation d/d, d 2 /d 2, d 3 /d 3,, or the prime notation,,,. Using the latter notation, the first two differential Jones & equations Bartlett in Learning, (2) can be written LLC a little more compactl as 6 e and NOT FOR 2 SALE 0, respectivel. OR DISTRIBUTION Actuall, the prime notation is used to denote onl the first three derivatives; the fourth derivative is written (4) instead of. In general, the nth derivative is d n /d n or (n). Although less convenient to write and to tpeset, the Leibniz notation has an advantage over the prime notation in that it clearl displas both the dependent and independent variables. For eample, in the differential equation d 2 /dt 2 6 0, it is immediatel Learning, seen that the smbol LLC now represents a dependent variable, & whereas Bartlett the Learning, independent LLC & Bartlett NOT FOR SALE variable OR is t. DISTRIBUTION You should also be aware that in phsical sciences NOT FOR and engineering, SALE OR Newton s DISTRIBUTIO dot notation (derogativel referred to b some as the flspeck notation) is sometimes used to denote derivatives with respect to time t. Thus the differential equation d 2 s/dt 2 32 becomes $ s 32. Partial derivatives are often denoted b a subscript notation indicating the independent variables. For eample, the first and second equations in (3) can be written, in turn, as & Bartlett Learning, u ullc 0 and u u tt u t. Classification b Order The order of a differential equation (ODE or PDE) is the order of the highest derivative in the equation. EXAMPLE 2 Order of a Differential Equation The differential equations highest order highest order T T d 2 3 5ad 2 d d b 2 4 e, 2 04 u 0 02 u & 4 0t Bartlett 0 Learning, LLC 2 are eamples of a second-order ordinar differential equation and a fourth-order partial differential equation, respectivel. A first-order ordinar differential equation is sometimes written in the differential form M(, ) d N(, ) d 0. EXAMPLE 3 Differential Form of a First-Order ODE If we assume that is the dependent variable in a first-order ODE, then recall from calculus that the differential d is defined to be d 9d. (a) B dividing NOT b the FOR differential SALE d OR an alternative DISTRIBUTION form of the equation ( 2 ) d 4 d 0 is given b d or equivalentl d 4 & Bartlett (b) B Learning, multipling the LLC differential equation d 6 d d. d NOT FOR SALE OR DISTRIBUTI b d we see that the equation has the alternative differential form ( 2 Jones 2 ) d & 6 Bartlett d 0. Learning, LLC In smbols, we can epress an nth-order ordinar differential equation in one dependent variable b the general form F(,,,, (n) ) 0, (4) where F is a real-valued function & Bartlett of n 2 variables: Learning,,, LLC,, (n). For both practical and theoretical reasons, NOT we shall FOR also SALE make the OR assumption DISTRIBUTION hereafter that it is possible to solve an. Definitions and Terminolog 5

6 & Bartlett Learning LLC, an Ascend Learning Compan.. ordinar differential equation in the form (4) uniquel for the highest derivative (n) in terms of the remaining n variables. The differential equation d n NOT FOR d n f (, SALE, 9, p OR, (n2) DISTRIBUTION ), (5) where f is a real-valued continuous function, is referred to as the normal form of (4). Thus, when it suits our purposes, we shall use the normal forms & Bartlett Learning, d d 2 LLC f (, ) and f (,, 9) 2 d d to represent general first- and second-order ordinar differential equations. EXAMPLE 4 Normal Form of an ODE (a) B solving for the derivative d/d the normal form of the first-order differential equation d d 4 is d d 2 4. (b) B solving for the derivative 0 the normal form of the second-order differential equation 6 0 is 6. Classification b Linearit An nth-order ordinar differential equation (4) is said to be linear in the variable if F is linear in,,, (n). This means that an nth-order ODE is linear & when Bartlett (4) is alearning, n () (n) a n2 LLC () ( n2) p a ()9 a 0 () Jones 2 g() & Bartlett 0 or Learning, LLC a n () d n d n a n2() d n2 d p a n2 () d d a 0() g(). (6) Two important special cases of (6) are linear first-order (n ) and linear second-order (n 2) ODEs. a () d d a 0() g() and anot 2 () d 2 FOR d a () SALE d 2 d OR a 0() DISTRIBUTION g(). (7) In the additive combination on the left-hand side of (6) we see that the characteristic two properties of a linear ODE are Remember these two characteristics of a The dependent variable and all its derivatives,,, (n) are of the first degree; that linear ODE. is, the power of each term involving is. The coefficients a 0, NOT a,, FOR a n of, SALE, OR, (n) DISTRIBUTION depend at most on the independent variable. A nonlinear ordinar differential equation is simpl one that is not linear. If the coefficients of,,, (n) contain the dependent variable or its derivatives or if powers of,,, (n), such & Bartlett as ( ) 2, appear Learning, the equation, LLC then the DE is nonlinear. Also, nonlinear & Bartlett functions Learning, LLC NOT FOR of the SALE dependent OR variable DISTRIBUTION or its derivatives, such as sin or NOT e cannot FOR appear SALE in a OR linear DISTRIBUTI equation. EXAMPLE 5 Linear and Nonlinear Differential Equations (a) The equations 6 CHAPTER Introduction to Differential Equations ( 2 ) d 4 d 0, , d 3 d e 3 d d are, in turn, eamples of linear first-, second-, and third-order ordinar differential equations. We have just demonstrated in part (a) & of Bartlett Eample 3 Learning, that the first LLC equation is linear in b writing it in the alternative NOT form FOR 4 SALE. OR DISTRIBUTION

7 & Bartlett Learning LLC, an Ascend Learning Compan.. (b) The equations nonlinear term: nonlinear term: nonlinear term: coefficient depends Jones & Bartlett nonlinear Learning, function of LLC power not NOT T FOR SALE OR DISTRIBUTIONT T d 2 d 4 ( 2 )9 2 e, sin 0, 2 d d 4 2 0, are eamples of nonlinear first-, second-, and fourth-order ordinar differential equations, & Bartlett respectivel. Learning, LLC Solution As stated before, one of our goals in this course is to solve or find solutions of differential equations. The concept of a solution of an ordinar differential equation is defined net. & Bartlett Learning, Definition LLC..2 Solution of an ODE An function f, defined on an interval NOT I and FOR possessing SALE at OR least DISTRIBUTION n derivatives that are continuous on I, which when substituted into an nth-order ordinar differential equation reduces the equation to an identit, is said to be a solution of the equation on the interval. In other words, a solution of an nth-order ordinar differential equation (4) is a function f that possesses at least n derivatives and F(, f(), f (),, f (n) ()) 0 for all in I. We sa that f satisfies the differential equation on I. For our purposes, we shall also assume that a solution f is a real-valued function. In our initial discussion we have alread seen that e 0.2 & Bartlett is a solution Learning, of d/d 0.2 LLCon the interval ( q, q). Occasionall it will be convenient to denote a solution b the alternative smbol (). Interval of Definition You can t think solution of an ordinar differential equation without simultaneousl thinking interval. The interval I in Definition..2 is variousl called the interval of definition, the interval of validit, or the domain of the solution and can be an open interval (a, b), a closed interval [a, b], an infinite interval (a, q), and so on. EXAMPLE 6 Verification of a Solution Verif that the indicated function is a solution of the given differential equation on the interval ( q, q). (a) d d >2 ; 6 4 & Bartlett (b) Learning, 2 LLC 0; e SOLUTION One wa of verifing that the given function is a solution is to see, after substituting, whether each side of the equation is the same for ever in the interval ( q, q). (a) From left-hand side: d d right-hand side: >2 a 4 6 b >2 NOT FOR 2 3SALE OR DISTRIBUTI 4 4, we see that each side of the equation is the same for ever real number. Note that /2 4 2 is, b definition, the nonnegative square root of 6 4. (b) From the derivatives e + e and e 2e we have for ever real number, left-hand side: 2 (e 2e ) 2(e e ) e 0 right-hand side: 0. Note, too, that in Eample 6 each differential equation possesses the constant solution 0, defined on ( q, q). A solution & of Bartlett a differential Learning, equation that LLC is identicall zero on an interval I is said to be a trivial NOT solution. FOR SALE OR DISTRIBUTION. Definitions and Terminolog 7

8 & Bartlett Learning LLC, an Ascend Learning Compan.. Solution Curve The graph of a solution f of an ODE is called a solution curve. Since f is a differentiable function, it is continuous on its interval I of definition. Thus there ma be a difference between the graph of Jones the function & Bartlett f and the graph Learning, of the solution LLC f. Put another wa, NOT FOR SALE OR DISTRIBUTION the domain of the function NOT f does FOR not need SALE to be OR the same DISTRIBUTION as the interval I of definition (or domain) of the solution f. EXAMPLE 7 Function vs. Solution (a) & Bartlett Considered Learning, simpl as a function, LLC the domain of / is the set of all real numbers NOT FOR ecept SALE 0. When OR we DISTRIBUTION graph /, we plot points in the -plane corresponding to a judicious (a) Function = /, 0 sampling of numbers taken from its domain. The rational function / is discontinuous at 0, and its graph, in a neighborhood of the origin, is given in FIGURE..(a). The function / is not differentiable at 0 since the -ais (whose equation is 0) is a vertical asmptote of the graph. & Bartlett Learning, LLC (b) Now / is also a solution of the linear first-order differential equation 0 (verif). But when we sa / is a solution of this DE we mean it is a function defined on an interval I on which it is differentiable and satisfies the equation. In other words, / is a solution of the DE on an interval not containing 0, such as ( 3, ), ( 2, 0), ( q, 0), or (0, q). Because the solution curves defined b / on the intervals ( 3, ) (b) Solution = /, (0, ) and on ( 2, 0) are simpl segments & Bartlett or pieces Learning, of the solution LLCcurves defined b FIGURE.. Eample 7 illustrates / on ( q, 0) and (0, q), respectivel, it makes sense to take the interval I to be as large the difference between the function as possible. Thus we would take I to be either ( q, 0) or (0, q). The solution curve on the / and the solution / interval (0, q) is shown in Figure..(b). Eplicit & Bartlett and Learning, Implicit Solutions LLC You should be familiar with the & terms Bartlett eplicit Learning, and LLC implicit functions from our stud of calculus. A solution in which the dependent variable is NOT FOR epressed SALE solel OR in terms DISTRIBUTION of the independent variable and constants NOT is said to FOR be an eplicit SALE solution. OR DISTRIBUTIO For our purposes, let us think of an eplicit solution as an eplicit formula f() that we can manipulate, evaluate, and differentiate using the standard rules. We have just seen in the last two eamples that 6 4, e, and / are, in turn, eplicit solutions of d/d /2, & Bartlett Learning, 2 LLC 0, and 0. Moreover, the Jones trivial solution & Bartlett 0 Learning, is an eplicit solution LLC of all three equations. We shall see when we get down to the business of actuall solving some ordinar differential equations that methods of solution do not alwas lead directl to an eplicit solution f(). This is particularl true when attempting to solve nonlinear first-order differential equations. Often we have to be content with a relation or epression G(, ) 0 that defines a solution f implicitl. Definition..3 Implicit NOT Solution FOR of SALE an ODE OR DISTRIBUTION A relation G(, ) 0 is said to be an implicit solution of an ordinar differential equation (4) on an interval I provided there eists at least one function f that satisfies the relation as well as the differential equation on I. NOT FOR It is SALE beond OR the scope DISTRIBUTION of this course to investigate the conditions NOT FOR under which SALE a relation OR DISTRIBUTI G(, ) 0 defines a differentiable function f. So we shall assume that if the formal implementation of a method of solution leads to a relation G(, ) 0, then there eists at least one function f that satisfies both the relation (that is, G(, f()) 0) and the differential equation on an interval I. If the implicit solution G(, ) 0 is fairl simple, we ma be able to solve for in terms & Bartlett of Learning, and obtain LLC one or more eplicit solutions. See Jones (iv) in the & Remarks. Bartlett Learning, LLC EXAMPLE 8 Verification of an Implicit Solution The relation is an implicit solution of the nonlinear differential equation & d Bartlett d Learning, LLC NOT FOR SALE OR DISTRIBUTION (8) 8 CHAPTER Introduction to Differential Equations

9 & Bartlett Learning LLC, an Ascend Learning Compan.. on the interval defined b 5 5. B implicit differentiation we obtain d d 2 d d 2 d d & Bartlett 25 Learning, or 2 LLC 2 0. (9) d d Solving the last equation in (9) for the smbol d/d gives (8). Moreover, solving for in terms of ields " The two functions f () " and f 2 () " satisf the relation (that is, 2 f 2 25 and 2 f ) and are eplicit solutions defined on the interval ( 5, 5). The solution curves given in FIGURE..2(b) and..2(c) are segments of the graph of the implicit solution in Figure..2(a) & Bartlett Learning, LLC (a) Implicit solution = 25 5 (b) Eplicit solution (c) Eplicit solution = 25 2, 5 < < 5 2 = 25 2, 5 < < 5 FIGURE..2 An implicit solution and two eplicit solutions in Eample 8 An relation of the form 2 2 c 0 formall satisfies (8) for an constant c. However, it is understood that the relation should alwas make sense in the real number sstem; thus, for eample, we cannot sa that is an implicit solution of the equation. Wh not? & Bartlett Because Learning, the distinction LLC between an eplicit solution and Jones an implicit & Bartlett solution should Learning, be intuitivel OR clear, DISTRIBUTION we will not belabor the issue b alwas NOT saing, FOR Here SALE is an eplicit OR DISTRIBUTIO (implicit) LLC NOT FOR SALE solution. Families of Solutions The stud of differential equations is similar to that of integral calculus. When evaluating an antiderivative or indefinite integral in calculus, we use a single constant & Bartlett Learning, LLC c of integration. Analogousl, when solving a first-order differential equation F(,, ) 0, we c > 0 usuall obtain a solution containing a NOT single arbitrar FOR SALE constant OR or parameter DISTRIBUTION c. A solution containing an arbitrar constant represents a set G(,, c) 0 of solutions called a one-parameter famil of solutions. When solving an nth-order differential equation F(,,,, (n) ) 0, we seek an n-parameter famil of solutions G(,, c, c 2,, c n ) 0. This means that a single c = 0 differential equation can possess an infinite number of solutions corresponding to the unlimited number of choices for the & parameter(s). Bartlett Learning, A solution of LLC a differential equation that is free c < 0 of arbitrar parameters NOT is FOR called a SALE particular OR solution. DISTRIBUTION For eample, the one-parameter famil FIGURE..3 Some solutions of c cos is an eplicit solution of the linear first-order equation 2 sin on the 2 sin interval ( q, q) (verif). FIGURE..3, obtained using graphing software, shows the graphs of some of the solutions in this famil. The solution cos, the red curve in the figure, is a particular solution corresponding to c 0. Similarl, on the interval ( q, q), c e c 2 e & Bartlett is a two-parameter Learning, famil LLC of solutions (verif) of the linear second-order Jones & equation Bartlett Learning, 2 0 LLC NOT FOR SALE in part OR (b) of DISTRIBUTION Eample 6. Some particular solutions of NOT the equation FOR SALE are the OR trivial DISTRIBUTI solution 0 (c c 2 0), e (c 0, c 2 ), 5e 2e (c 5, c 2 2), and so on. In all the preceding eamples, we have used and to denote the independent and dependent variables, respectivel. But ou should become accustomed to seeing and working with other smbols to denote these variables. For eample, we could denote the independent variable b t & Bartlett Learning, and the LLC dependent variable b. EXAMPLE 9 Using Different Smbols The functions c cos 4t and c 2 sin 4t, where c and c 2 are arbitrar constants or parameters, are both solutions of the linear differential equation NOT FOR SALE OR 6 DISTRIBUTION 0.. Definitions and Terminolog 9

10 & Bartlett Learning LLC, an Ascend Learning Compan.. For c cos 4t, the first two derivatives with respect to t are 4c sin 4t and 6c cos 4t. Substituting and then gives Jones 6 6c & Bartlett cos 4t 6(c Learning, cos 4t) LLC 0. In like manner, for c 2 sin 4t we have 6c 2 sin 4t, and so 0 CHAPTER Introduction to Differential Equations 6 6c 2 sin 4t 6(c 2 sin 4t) 0. Finall, it is straightforward to verif that the linear combination of solutions for the twoparameter & Bartlett famil Learning, c cos 4t LLC c 2 sin 4t is also a solution of the Jones differential & Bartlett equation. Learning, LLC The net eample shows that a solution of a differential equation can be a piecewise-defined function. EXAMPLE 0 A Piecewise-Defined Solution c = & Bartlett Learning, You should verif LLCthat the one-parameter famil c 4 is & a Bartlett one-parameter Learning, famil of solutions LLC NOT FOR SALE OR DISTRIBUTION of the linear differential equation 4 NOT 0 on FOR the interval SALE ( q, OR q). DISTRIBUTION See FIGURE..4(a). The piecewise-defined differentiable function c = e 4,, 0 (a) 4, $ 0 is a particular solution of the equation but cannot be obtained from the famil c 4 b a c = single choice of c; the solution NOT is FOR constructed SALE from OR the DISTRIBUTION famil b choosing c for 0 0 and c for 0. See Figure..4(b). c = Singular Solution Sometimes a differential equation possesses a solution that is not a < 0 member of a famil of solutions of the equation; that is, a solution that cannot be obtained b specializing & Bartlett an of Learning, the parameters LLC in the famil of solutions. Such an etra solution & Bartlett is called Learning, a LLC (b) NOT FOR singular SALE solution. OR For DISTRIBUTION eample, we have seen that 6 4 and NOT 0 FOR are solutions SALE of OR the differential equation d/d /2 on ( q, q). In Section 2.2 we shall demonstrate, b actuall DISTRIBUTIO FIGURE..4 Some solutions of 4 0 in Eample 0 solving it, that the differential equation d/d /2 possesses the one-parameter famil of solutions ( 4 2 c) 2, c 0. When c 0, the resulting particular solution is 6 4. But notice that the trivial solution 0 is a singular solution since it is not a member of the famil & Bartlett Learning, ( LLC 4 2 c) 2 ; there is no wa of assigning a value to the constant c to obtain 0. Sstems of Differential Equations Up to this point we have been discussing single differential equations containing one unknown function. But often in theor, as well as in man applications, we must deal with sstems of differential equations. A sstem of ordinar differential equations is two or more equations involving the derivatives of two or more unknown functions of a single independent variable. & Bartlett For eample, Learning, if and denote LLCdependent variables and t the independent variable, NOT then FOR a sstem SALE of two OR first-order DISTRIBUTION differential equations is given b d f (t,, ) dt (0) d g(t,, ). dt NOT FOR SALE OR DISTRIBUTI A solution of a sstem such as (0) is a pair of differentiable functions f (t), f 2 (t) defined on a common interval I that satisf each equation of the sstem on this interval. See Problems 4 and 42 in Eercises.. NOT FOR SALE OR REMARKS DISTRIBUTION (i) It might not be apparent whether a first-order ODE written in differential form M(, ) d N(, ) d 0 is linear or nonlinear because there is nothing in this form that tells us which smbol denotes the dependent variable. See Problems 9 and 0 in Eercises.. (ii) We will see in the chapters that follow that a solution of a differential equation ma involve an integral-defined function. One wa of defining a function F of a single variable b

11 & Bartlett Learning LLC, an Ascend Learning Compan.. means of a definite integral is & Bartlett Learning, F() # LLC g(t) dt. () a If the integrand g in () is continuous on an interval [a, b] and a b, then the derivative form of the Fundamental Theorem of Calculus states that F is differentiable on (a, b) and F9() d# d g(t) dt Jones g(). & Bartlett Learning, (2) LLC a The integral in () is often nonelementar, that is, an integral of a function g that does not have an elementar-function antiderivative. Elementar functions include the familiar functions studied in a tpical precalculus course: constant, polnomial, rational, eponential, logarithmic, trigonometric, and inverse & Bartlett Learning, trigonometric LLC functions, as well as rational powers of these functions, finite combinations of these functions using addition, subtraction, multiplication, division, and function compositions. For eample, even though e t 2, " t 3, and cos t 2 are elementar functions, the integrals ee t 2 dt, e" t 3 dt, and ecos t 2 dt are nonelementar. See Problems in Eercises.. (iii) Although the concept Jones of a & solution Bartlett of a differential Learning, equation LLChas been emphasized in this section, ou should NOT be aware FOR that SALE a DE does OR not DISTRIBUTION necessaril have to possess a solution. See Problem 43 in Eercises.. The question of whether a solution eists will be touched on in the net section. (iv) A few last words about implicit solutions of differential equations are in order. In Eample 8 we were able to solve the relation for in terms of to get two eplicit solutions, f () " and f 2 () "25 2 2, of the differential equation (8). But don t NOT FOR SALE read OR too much DISTRIBUTION into this one eample. Unless it is eas, NOT obvious, FOR or important, SALE OR or ou DISTRIBUTIO are instructed to, there is usuall no need to tr to solve an implicit solution G(, ) 0 for eplicitl in terms of. Also do not misinterpret the second sentence following Definition..3. An implicit solution G(, ) 0 can define a perfectl good differentiable function f that is a solution of a DE, but et we ma not be able to solve G(, ) 0 using analtical methods such as algebra. The solution curve of f ma be a segment or piece of the graph of G(, ) 0. See Problems 49 and 50 in Eercises NOT.. FOR SALE OR DISTRIBUTION (v) If ever solution of an nth-order ODE F(,,,, (n) ) 0 on an interval I can be obtained from an n-parameter famil G(,, c, c 2,, c n ) 0 b appropriate choices of the parameters c i, i, 2,, n, we then sa that the famil is the general solution of the DE. In solving linear ODEs, we shall impose relativel simple restrictions on the coefficients of the equation; with these restrictions can & be Bartlett assured that Learning, not onl does LLC a solution eist on an interval but also that a famil NOT of solutions FOR ields SALE all OR possible DISTRIBUTION solutions. Nonlinear equations, with the eception of some first-order DEs, are usuall difficult or even impossible to solve in terms of familiar elementar functions. Furthermore, if we happen to obtain a famil of solutions for a nonlinear equation, it is not evident whether this famil contains all solutions. On a practical level, then, the designation general solution is applied onl to linear DEs. Don t & Bartlett be concerned Learning, about this LLC concept at this point but store the Jones words general & Bartlett solution Learning, the back LLC NOT FOR SALE of our OR mind we DISTRIBUTION will come back to this notion in Section NOT 2.3 FOR and again SALE in Chapter OR DISTRIBUTI 3.. Eercises Answers to selected odd-numbered problems begin on page ANS-. In Problems NOT FOR 8, state SALE the order OR of DISTRIBUTION the given ordinar 3. t 5 NOT (4) t FOR 3 6 SALE 0 OR DISTRIBUTION differential equation. Determine whether the equation is linear or nonlinear b matching it with (6). 4. d 2 u du u cos (r u) 2. ( ) 4 5 cos dr dr d d 2 ad 3 d b 0 5. d 2 2 & Bartlett d Learning, ad 2 Å d b LLC. Definitions and Terminolog

12 & Bartlett Learning LLC, an Ascend Learning Compan.. 6. d 2 R dt k 2 R 2 7. (sin u) & Bartlett (cos u) Learning, 2 LLC NOT 8. FOR $ 2 (SALE 2 3 # 2 ) OR # DISTRIBUTION 0 In Problems 9 and 0, determine whether the given first-order differential equation is linear in the indicated dependent variable b matching it with the first differential equation given in (7). 9. ( 2 ) d d 0; in ; in 0. u dv (v uv ue u ) du 0; in v; in u In Problems 4, verif that the indicated function is an eplicit solution of Jones the given & differential Bartlett equation. Learning, Assume LLC an appropriate interval I of definition for each solution.. 2 0; e /2 2. d 20 24; dt e 20t 3. & 6 Bartlett 3 0; Learning, e 3 cos LLC 2 NOT 4. FOR SALE tan ; OR DISTRIBUTION (cos ) ln(sec tan ) In Problems 5 8, verif that the indicated function f() is an eplicit solution of the given first-order differential equation. Proceed as in Eample 7, b considering f simpl as a function, give its domain. Then b considering f as a solution of the differential equation, give at least one interval I of definition. 5. ( 2 )9 2 8; 4" ; 5 tan ; /(4 2 ) cos ; ( sin ) NOT FOR SALE /2 OR DISTRIBUTION In Problems 9 and 20, verif that the indicated epression is an implicit solution of the given first-order differential equation. Find at least one eplicit solution f() in each case. Use a graphing utilit to obtain the graph of an eplicit solution. Give an interval & Bartlett I of definition Learning, of each LLC solution f. In Problems 35 and 36, find values of m so that the function Jones m is a & solution Bartlett of the Learning, given differential LLCequation. NOT FOR 9. dx SALE OR DISTRIBUTION 2 NOT 35. FOR 2 SALE 0 OR DISTRIBUTION (X 2 )( 2 2X); lna2x dt X 2 b t In Problems 37 40, use the concept that c, q q, d ( 2 ) d 0; is a constant function if and onl if 0 to determine whether the given differential equation possesses constant solutions. In Problems 2 24, verif that the indicated Jones famil & Bartlett of functions Learning, 37. LLC Jones 2 2 & Bartlett 3 Learning, LLC is a solution of the given differential NOT equation. FOR Assume SALE an OR DISTRIBUTION 39. appropriate interval I of definition for each solution. ( ) 40. NOT 4 FOR 6 SALE 0 OR DISTRIBUTI 2. dp dt P( 2 P); P c e t In Problems 4 and 42, verif that the indicated pair of functions c e t is a solution of the given sstem of differential equations on the 22. d d 4 interval ( q, q). 83 ; & Bartlett Learning, c e LLC d 2 d 2 4 d 4. d 3 4 0; 2 d c dt 42. d 2 NOT FOR SALE OR 4DISTRIBUTION dt 2 et e 2 c 2 e 2 d d d 3 3 d d d 2 d 5 3; 2 d 2 dt dt et ; 2 ; e& 2t Bartlett 3e 6t, Learning, LLC cos 2t sin 2t 5 et, NOT FOR c SALE c 2 OR c 3 DISTRIBUTION ln 4 2 NOT FOR esale 2t 5eOR 6t DISTRIBUTION cos 2t sin 2t 5 et 2 CHAPTER Introduction to Differential Equations In Problems 25 28, use (2) to verif that the indicated function is a solution of the given differential equation. Assume an appropriate Jones & interval Bartlett I of definition Learning, of each LLC solution. NOT FOR d 25. d 2 SALE 3 ; OR DISTRIBUTION # e 3t e3 dt t d d 2 2 cos ; " # cos t 4 "t dt 27. d 2 d 0 sin ; 5 0 # sin t NOT FOR SALE dt OR DISTRIBUTIO t 28. d 2 ; e 2 e 2 d # e t 2 dt Verif that the piecewise-defined & Bartlett function Learning, LLC NOT FOR SALE e OR DISTRIBUTION 2,, 0 2, $ 0 is a solution of the differential equation 2 0 on the interval ( q, q). NOT 30. In FOR Eample SALE 8 we OR saw DISTRIBUTION that f () " and f 2 () " are solutions of d/d / on the interval ( 5, 5). Eplain wh the piecewise-defined function e "25 2 2, Jones 5, &, Bartlett 0 Learning, LLC "25 2 NOT 2, FOR 0 # SALE, 5 OR DISTRIBUTIO is not a solution of the differential equation on the interval ( 5, 5). In Problems 3 34, find values of m so that the function e m is a solution of the given differential & Bartlett equation. Learning, LLC NOT FOR SALE OR 4DISTRIBUTION

13 & Bartlett Learning LLC, an Ascend Learning Compan.. Discussion Problems an interval I of definition of a solution f of the DE? Carr out our ideas and compare with our estimates of the intervals in 43. Make up a differential equation that does not possess an real & Bartlett solutions. Learning, LLC & Bartlett Problem 50. Learning, LLC NOT FOR 53. SALE In Eample OR 8, DISTRIBUTION the largest interval I over which the eplicit 44. Make up a differential equation that ou feel confident possesses onl the trivial solution 0. Eplain our reasoning. solutions f () and f 2 () are defined is the open interval ( 5, 5). Wh can t the interval I of definition be the 45. What function do ou know from calculus is such that its first closed interval [ 5, 5]? derivative is itself? Its first derivative is a constant multiple k 54. In Problem 2, a one-parameter famil of solutions of the DE of itself? Write each answer Jones in & the Bartlett form of a first-order Learning, dif-llferential equation with NOT a solution. FOR SALE OR DISTRIBUTION the point (0, 3)? Through NOT the FOR point SALE (0, )? OR DISTRIBUTIO P P( P) is given. Does an & solution Bartlett curve Learning, pass through LLC 46. What function (or functions) do ou know from calculus is 55. Discuss, and illustrate with eamples, how to solve differential equations of the forms d/d f () and d 2 /d 2 f (). such that its second derivative is itself? Its second derivative is the negative of itself? Write each answer in the form of a 56. The differential equation ( ) second-order differential equation with a solution has the form given in (4). Determine & Bartlett whether Learning, the equation LLC can be put into 47. Given that sin is an eplicit solution of the first-order the NOT normal FOR form SALE d/d OR f (, ). DISTRIBUTION differential equation d/d " 2 2. Find an interval I of 57. The normal form (5) of an nth-order differential equation definition. [Hint: I is not the interval ( q, q).] is equivalent to (4) whenever both forms have eactl the 48. Discuss wh it makes intuitive sense to presume that the linear differential equation sin t has a solution for which F(,, ) 0 is not equivalent to the normal form same solutions. Make up a first-order differential equation & Bartlett of the form Learning, A sin t LLC B cos t, where A and B are constants. & Bartlett d/d f (, Learning, ). LLC NOT FOR SALE Then find OR specific DISTRIBUTION constants A and B so that A sin t B NOT cos t FOR 58. SALE Find a linear OR second-order DISTRIBUTION differential equation F(,,, ) 0 is a particular solution of the DE. for which c c 2 2 is a two-parameter famil of solutions. Make sure that our equation is free of the arbitrar In Problems 49 and 50, the given figure represents the graph parameters c and c 2. of an implicit solution G(, ) 0 of a differential equation d/d f (, ). In each case Jones the relation & Bartlett G(, ) Learning, 0 implicitl LLCQualitative information about a solution & Bartlett f() of Learning, a LLC defines several solutions NOT of the FOR DE. Carefull SALE reproduce OR DISTRIBUTION each differential equation can NOT often be FOR obtained SALE from OR the equation DISTRIBUTIO figure on a piece of paper. Use different colored pencils to mark itself. Before working Problems 59 62, recall the geometric off segments, or pieces, on each graph that correspond to graphs significance of the derivatives d/d and d 2 /d 2. of solutions. Keep in mind that a solution f must be a function 59. Consider the differential equation d/d e 2. and differentiable. Use the solution curve to estimate the (a) Eplain wh a solution of the DE must be an increasing interval I Jones of definition & Bartlett of each solution Learning, f. LLC function Jones on & an Bartlett interval of Learning, the -ais. LLC 49. NOT FOR SALE OR 50. DISTRIBUTION (b) NOT What FOR are SALE lim d/d OR and DISTRIBUTION limd/d? What does this S 2q Sq suggest about a solution curve as S q? (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. & Bartlett (d) Sketch Learning, the graph of a LLC solution f() of the differential NOT FOR SALE equation OR DISTRIBUTION whose shape is suggested b parts (a) (c). 60. Consider the differential equation d/d 5. (a) Either b inspection, or b the method suggested in Problems 37 40, find a constant solution of the DE. FIGURE..5 Graph for Problem 49 FIGURE..6 Graph for Problem The graphs of the members of the one-parameter famil 3 3 3c are called folia of Descartes. Verif that this famil is an implicit solution of the first-order differential equation & Bartlett d Learning, d ( ) LLC NOT FOR SALE OR (2 DISTRIBUTION ). 52. The graph in FIGURE..6 is the member of the famil of folia in Problem 5 corresponding to c. Discuss: How can the DE in Problem 5 help in finding points on the graph of where the tangent line is vertical? How does knowing where a tangent line is vertical help in determining (b) Using onl the differential equation, find intervals on the -ais on which a nonconstant Jones & Bartlett solution Learning, f() is increasing. Find intervals NOT FOR on the SALE -ais on OR which DISTRIBUTI f() LLC is decreasing. 6. Consider the differential equation d/d (a b), where a and b are positive constants. (a) Either b inspection, or b the method suggested in Problems 37 40, find two constant solutions of the DE. (b) Using onl the differential equation, find intervals on the -ais on which a nonconstant solution f() is increasing. On which f() is decreasing. (c) Using onl the differential equation, eplain wh a/2b & Bartlett is the -coordinate Learning, of LLC a point of inflection of the graph of NOT FOR SALE a nonconstant OR DISTRIBUTION solution f().. Definitions and Terminolog 3

14 & Bartlett Learning LLC, an Ascend Learning Compan.. (d) On the same coordinate aes, sketch the graphs of the two Computer Lab Assignments constant solutions found in part (a). These constant solutions & Bartlett partition the Learning, -plane into LLC three regions. In each re- In Problems 63 and 64, use a CAS to compute all derivatives and to carr out the simplifications needed to verif that the NOT FOR gion, SALE sketch OR the graph DISTRIBUTION of a nonconstant solution f() NOT indicated FOR function SALE is OR a particular DISTRIBUTION solution of the given differential equation. whose shape is suggested b the results in parts (b) and (c). 62. Consider the differential equation (4) ; (a) Eplain wh there eist no constant solutions of the DE. e 5 cos 2 (b) Describe the graph of a solution f(). For eample, can a solution curve have an relative etrema? ; (c) Eplain wh 0 is the -coordinate of a point of inflection of a solution curve cos (5 ln ) sin (5 ln ) (d) Sketch the graph of a solution f() of the differential equation whose shape is suggested b parts (a) (c). 4 CHAPTER Introduction to Differential Equations.2 Initial-Value Problems INTRODUCTION We are NOT often interested FOR SALE in problems OR DISTRIBUTION in which we seek a solution () of a differential equation so that () satisfies prescribed side conditions that is, conditions that are imposed on the unknown () or on its derivatives. In this section we eamine one such problem called an initial-value problem. Initial-Value & Bartlett Learning, Problem LLC On some interval I containing 0, the problem d n Solve: d n f (,, 9, p, (n2) ) () Subject to: ( 0 ) 0, 9( 0 ), p, (n2) ( 0 ) n2, & Bartlett where Learning, 0,,, LLC n are arbitraril specified real constants, Jones is called & Bartlett an initial-value Learning, problem LLC (IVP). NOT FOR SALE OR The DISTRIBUTION values of () and its first n derivatives NOT at a single FOR point SALE 0 : ( OR 0 ) DISTRIBUTION 0, ( 0 ),, (n ) ( 0 ) n, are called initial conditions (IC). First- and Second-Order IVPs The problem given in () is also called an nth-order initial-value problem. For eample, & d Bartlett Learning, LLC solutions of the DE NOT FOR SALE OR DISTRIBUTION NOT Solve: FOR SALE f (, ) d OR DISTRIBUTION (2) Subject to: ( 0 ) 0 ( d 2 0, 0 ) and Solve: f (,, 9) 2 d & Bartlett Learning, (3) LLC I Subject to: ( 0 ) 0, 9( 0 ) NOT FOR SALE OR DISTRIBUTI FIGURE.2. First-order IVP are first- and second-order initial-value problems, respectivel. These two problems are eas to interpret in geometric terms. For (2) we are seeking a solution of the differential equation on solutions of the DE an interval I containing 0 so that a solution curve passes through the prescribed point ( 0, 0 ). & Bartlett See Learning, FIGURE.2.. LLC For (3) we want to find a solution of the differential & Bartlett equation Learning, whose graph LLCnot NOT FOR SALE OR onl DISTRIBUTION passes through ( 0, 0 ) but passes through NOT so that FOR the slope SALE of the OR curve DISTRIBUTION at this point is. m = See FIGURE.2.2. The term initial condition derives from phsical sstems where the independent variable is time t and where (t 0 ) 0 and (t 0 ) represent, respectivel, the position and ( 0, 0 ) velocit of an object at some beginning, or initial, time t 0. Solving an nth-order initial-value problem frequentl entails using an n-parameter famil of I solutions of the given differential equation & Bartlett to find n specialized Learning, constants LLCso that the resulting NOT FIGURE FOR.2.2 SALE Second-order OR DISTRIBUTION IVP particular solution of the equation NOT also FOR fits that SALE OR is, satisfies the DISTRIBUTION n initial conditions.

15 & Bartlett Learning LLC, an Ascend Learning Compan.. (0, 3) EXAMPLE First-Order IVPs (a) It is readil verified that ce is a one-parameter famil of solutions of the simple first-order equation on & the Bartlett interval ( q, Learning, q). If we LLC specif an initial condition, sa, (0) 3, then substituting NOT FOR SALE 0, 3 OR in the DISTRIBUTION famil determines the constant 3 ce 0 c. Thus the function 3e is a solution of the initial-value problem, (0) 3. (b) Now if we demand that a solution of the differential equation pass through the point (, 2) & Bartlett Learning, LLC (, 2) rather than (0, 3), then () 2 will ield 2 ce or c 2e. The function NOT FOR SALE OR 2e DISTRIBUTION is a solution of the initial-value problem, () 2. FIGURE.2.3 Solutions of IVPs in Eample The graphs of these two solutions are shown in blue in FIGURE.2.3. & Bartlett Learning, The LLC net eample illustrates another first-order Jones & initial-value Bartlett problem. Learning, In this LLC eample, notice how the interval I of definition of the NOT solution FOR () depends SALE on OR the DISTRIBUTION initial condition ( 0 ) 0. EXAMPLE 2 Interval I of Definition of a Solution In Problem 6 of Eercises 2.2 ou will be asked to show that a one-parameter famil of solutions of the first-order differential equation is /( 2 c). If we impose the initial condition (0), then substituting 0 and into the famil of solutions gives /c or c NOT. Thus, FOR SALE /( 2 ). OR We DISTRIBUTION now emphasize the following three distinctions. Considered as a function, the domain of /( 2 ) is the set of real numbers for which () is defined; this is the set of all real numbers ecept and. See FIGURE.2.4(a). & Bartlett Considered Learning, as a solution LLC of the differential equation & 2 Bartlett 2 0, the Learning, interval I LLC of definition of /( 2 ) could be taken to be an interval over which () is (a) Function defined for NOT all FOR SALE OR DISTRIBUTION defined and differentiable. As can be seen in Figure.2.4(a), the largest intervals on which ecept = ± /( 2 ) is a solution are ( q, ), (, ), and (, q). Considered as a solution of the initial-value problem 2 2 0, (0), the interval I of definition of /( 2 ) could be taken to be an interval over which () is defined, & Bartlett Learning, LLC differentiable, and contains the initial Jones point & Bartlett 0; the largest Learning, interval for which LLC this is true is (, ). See Figure.2.4(b). (0, ) See Problems 3 6 in Eercises.2 for a continuation of Eample 2. & Bartlett (b) Solution Learning, defined on interval LLC NOT FOR SALE containing OR DISTRIBUTION = 0 FIGURE.2.4 Graphs of function and solution of IVP in Eample 2 EXAMPLE 3 Second-Order IVP In Eample 9 of Section Jones. & we Bartlett saw that Learning, c cos 4t LLC c 2 sin 4t is a two-parameter famil of solutions of 6 0. Find a solution of the initial-value problem 0 6 0, (p/2) 2, (p/2). (4) SOLUTION We first appl (p/2) 2 to the given famil of solutions: c cos 2p c 2 sin 2p 2. Since cos 2p and sin 2p 0, we find that c 2. We net appl (p/2) to & Bartlett the one-parameter Learning, famil LLC(t) 2 cos 4t c 2 sin 4t. Differentiating & Bartlett and Learning, then setting LLC NOT FOR SALE t OR p/2 and DISTRIBUTION gives 8 sin 2p 4c 2 cos 2p, NOT from FOR which we SALE see that OR c 2 DISTRIBUTI 4. Hence 2 cos 4t 4 sin 4t is a solution of (4). Eistence and Uniqueness Two fundamental questions arise in considering an initialvalue problem: Does a solution of the problem eist? If a solution eists, is it unique? For a first-order initial-value problem such as (2), we ask: Does the differential equation d/d f (, ) possess solutions? Eistence Do an of the solution curves pass through the point ( 0, 0 )? Uniqueness When can we be & certain Bartlett that there Learning, is precisel LLC one solution curve passing through the NOT point FOR ( 0, 0 )? SALE OR DISTRIBUTION.2 Initial-Value Problems 5

16 & Bartlett Learning LLC, an Ascend Learning Compan.. Note that in Eamples and 3, the phrase a solution is used rather than the solution of the problem. The indefinite article a is used deliberatel to suggest the possibilit that other solutions ma eist. At this point it has not been demonstrated that there is a single solution of each & Bartlett Learning, LLC = 4 /6 problem. The net eample NOT illustrates FOR an initial-value SALE OR problem DISTRIBUTION with two solutions. = 0 (0, 0) FIGURE.2.5 Two solutions of the same IVP in Eample 4 EXAMPLE 4 An IVP Can Have Several Solutions Each of the functions 0 and 6 4 satisfies the differential equation d/d /2 and the initial condition (0) 0, and so the initial-value problem d/d /2, (0) 0, has at least two solutions. As illustrated in FIGURE.2.5, the graphs of both functions pass through the same point (0, 0). Within the safe confines of a formal course in differential equations one can be fairl confident that most differential equations will have solutions and that solutions of initial-value & Bartlett problems Learning, will probabl LLC be unique. Real life, however, Jones is not & Bartlett so idllic. Thus Learning, it is desirable LLCto know in advance of tring to solve an initial-value problem whether a solution eists and, when it does, whether it is the onl solution of the problem. Since we are going to consider firstorder differential equations in the net two chapters, we state here without proof a straightforward theorem that gives conditions that are sufficient to guarantee the eistence and uniqueness of a solution of a first-order initial-value problem of the form given in (2). We shall wait until Chapter 3 to address Jones the question & Bartlett of eistence Learning, and uniqueness LLC of a second-order initial-value problem. d Theorem.2. Eistence of a Unique Solution R Let R be a rectangular region in the -plane defined b a b, c d, that contains the point ( 0, 0 ) in its interior. If f (, ) and f/ are continuous on R, then there eists some ( 0, 0 ) interval I 0 : ( 0 h, 0 h), h 0, contained in [a, b], and a unique function () defined on NOT FOR I 0 that SALE is a solution OR DISTRIBUTION of the initial-value problem (2). c The foregoing result is one of the most popular eistence and uniqueness theorems for firstorder differential equations, because the criteria of continuit of f (, ) and f/ are relativel a I 0 b FIGURE.2.6 Rectangular region R eas to check. The geometr of Theorem.2. is illustrated in FIGURE.2.6. EXAMPLE 5 Eample 4 Revisited We saw in Eample 4 that the differential equation d/d /2 possesses at least two solutions whose graphs pass through (0, 0). Inspection of the functions 0f f (, ) & Bartlett >2 and Learning, 0 LLC 2 >2 shows that the are continuous in the upper half-plane defined b 0. Hence Theorem.2. enables us to conclude that through an point ( 0, 0 ), 0 0, in the upper half-plane there is some interval centered at 0 on which the given differential equation has a unique solution. Thus, for eample, even without solving it we know that there eists some interval & Bartlett centered Learning, at 2 on which LLC the initial-value problem d/d /2 &, (2) Bartlett, has Learning, a LLC NOT FOR unique SALE solution. OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTI In Eample, Theorem.2. guarantees that there are no other solutions of the initial-value problems, (0) 3, and, () 2, other than 3e and 2e, respectivel. This follows from the fact that f (, ) and f/ are continuous throughout the & Bartlett entire Learning, -plane. LLC It can be further shown that the Jones interval & I on Bartlett which each Learning, solution is LLC defined NOT FOR SALE OR is DISTRIBUTION ( q, q). Interval of Eistence/Uniqueness Suppose () represents a solution of the initial-value problem (2). The following three sets on the real -ais ma not be the same: the domain of the function (), Jones the interval & Bartlett I over which Learning, the solution LLC() is defined or eists, and the interval I 0 of eistence NOT FOR and uniqueness. SALE OR In Eample DISTRIBUTION 7 of Section. we illustrated 6 CHAPTER Introduction to Differential Equations

17 & Bartlett Learning LLC, an Ascend Learning Compan.. the difference between the domain of a function and the interval I of definition. Now suppose ( 0, 0 ) is a point in the interior of the rectangular region R in Theorem.2.. It turns out that the continuit of the function f (, ) & on Bartlett R b itself Learning, is sufficient to LLC guarantee the eistence of at least one solution of d/d NOT f (, FOR ), ( SALE 0 ) = 0, defined OR DISTRIBUTION on some interval I. The interval I of definition for this initial-value problem is usuall taken to be the largest interval containing 0 over which the solution () is defined and differentiable. The interval I depends on both f (, ) and the initial condition ( 0 ) 0. See Problems 3 34 in Eercises.2. The etra condition of continuit of the first partial derivative f/ on R enables us to sa that not onl does a solution eist & Bartlett on some interval Learning, I 0 containing LLC 0, but it also is the onl solution Jones satisfing & Bartlett ( 0 ) Learning, 0. However, LLC NOT FOR SALE Theorem OR.2. DISTRIBUTION does not give an indication of the sizes NOT of the intervals FOR SALE I and I 0 OR ; the interval DISTRIBUTIO I of definition need not be as wide as the region R and the interval I 0 of eistence and uniqueness ma not be as large as I. The number h 0 that defines the interval I 0 : ( 0 h, 0 h), could be ver small, and so it is best to think that the solution () is unique in a local sense, that is, a solution defined near the point ( 0, 0 ). See Problem 50 in Eercises.2. REMARKS.2 Eercises In Problems and 2, /( c e ) is a one-parameter famil 9. (p/6) 2, (p/6) 0 & of solutions Bartlett of Learning, the first-order LLC DE 2. Find a solution of Jones 0. & Bartlett (p/4)!2, Learning, (p/4) LLC 2!2 the first-order IVP consisting of this differential equation and NOT FOR the given initial condition. In Problems SALE OR 4, DISTRIBUTION c e c 2 e is a two-parameter famil. (0) of solutions of the second-order DE 0. Find a solution 3 2. ( ) 2 of the second-order IVP consisting of this differential equation In Problems 3 6, /( 2 c) is a one-parameter famil of and the given initial conditions. solutions of the first-order DE 2 & Bartlett 2 0. Find Learning, a solution LLC. (0), (0) 2 2. & () Bartlett 0, () Learning, e LLC of the first-order IVP consisting NOT FOR of this SALE differential OR equation DISTRIBUTION and 3. ( ) 5, ( ) NOT 5 FOR 4. SALE (0) 0, OR (0) DISTRIBUTI 0 the given initial condition. Give the largest interval I over which In Problems 5 and 6, determine b inspection at least two the solution is defined. solutions of the given first-order IVP. 3. (2) 3 4. ( 2) /3, (0) , (0) 0 5. (0) & Bartlett Learning, 6. ( 2) 4LLC In Problems 7 24, & determine Bartlett a region Learning, of the -plane LLC for which In Problems NOT FOR 7 0, SALE c cos OR t DISTRIBUTION c 2 sin t is a two-parameter famil of solutions of the second-order DE 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. 7. (0), (0) 8 8. (p/2) 0, (p/2) (i) The conditions in Theorem.2. are sufficient but not necessar. When f (, ) and f/ are continuous on a rectangular region R, it must alwas follow that a solution of (2) eists and is unique whenever ( 0, & 0 ) Bartlett is a point interior Learning, to R. However, LLC if the conditions stated in the hpotheses NOT of Theorem FOR.2. SALE do not OR hold, DISTRIBUTION then anthing could happen: Problem (2) ma still have a solution and this solution ma be unique, or (2) ma have several solutions, or it ma have no solution at all. A rereading of Eample 4 reveals that the hpotheses of Theorem.2. do not hold on the line 0 for the differential equation d/d /2, and so it is not surprising, as we saw in Eample 4 of this section, that there are two solutions defined on a common interval ( h, h) satisfing (0) 0. Jones On the other & Bartlett hand, the Learning, hpotheses LLC NOT FOR SALE of Theorem OR DISTRIBUTION.2. do not hold on the line for the NOT differential FOR equation SALE d/d OR DISTRIBUTIO. Nevertheless, it can be proved that the solution of the initial-value problem d/d, (0), is unique. Can ou guess this solution? (ii) You are encouraged to read, think about, work, and then keep in mind Problem 49 in Eercises.2. Answers to selected odd-numbered problems begin on page ANS-. the given NOT differential FOR SALE equation OR would DISTRIBUTION have a unique solution whose graph passes through a point ( 0, 0 ) in the region. 7. d 2/3 8. d! d d 9. & Bartlett d d Learning, LLC 20. d d 2.2 Initial-Value Problems 7

18 & Bartlett Learning LLC, an Ascend Learning Compan.. 2. (4 2 2 ) ( 3 )9 2 (c) The point ( 2, 3) is on the graph of , but 23. ( 2 2 ) ( 2 )9 which of the eplicit solutions in part (b) satisfies ( 2) & Bartlett 3? Learning, LLC In Problems 25 28, determine whether Theorem.2. guarantees that the differential equation 9 " possesses a find an implicit solution of the initial-value problem NOT 34. (a) FOR Use SALE the famil OR of DISTRIBUTION solutions in part (a) of Problem 33 to unique solution through the given point. d/d 3, (2) 4. Then, b hand, sketch the graph 25. (, 4) 26. (5, 3) of the eplicit solution of this problem and give its interval I of definition. 27. (2, 3) 28. (, ) 29. (a) B inspection, find a one-parameter famil of solutions (b) Are there an eplicit solutions of d/d & Bartlett 3 that pass Learning, LLC of the differential equation. Verif that each member of the famil is a solution of the initial-value problem through the origin? In Problems 35 38, the graph of a member of a famil of solutions of a second-order differential equation d 2 /d 2 f (,, ), (0) 0. (b) Eplain part (a) b determining a region R in the -plane is given. Match the solution curve with at least one pair of the for which the differential & Bartlett equation Learning, would LLC have a following initial conditions. Jones & Bartlett Learning, LLC unique NOT solution FOR through SALE a point OR ( 0 DISTRIBUTION, 0 ) in R. (a) () NOT, () FOR 2 SALE (b) ( ) OR DISTRIBUTION 0, ( ) 4 (c) Verif that the piecewise-defined function (c) (), () 2 (d) (0), (0) 2 (e) (0), (0) 0 (f) (0) 4, (0) 2 e 0,, 0, $ 0 satisfies the condition (0) 0. Determine whether this NOT FOR function SALE is OR also a DISTRIBUTION solution of the initial-value problem in part (a). 30. (a) Verif that tan ( c) is a one-parameter famil of solutions of the differential equation 2. (b) Since f (, ) 2 and f/ 2 are continuous 5 everwhere, the region R NOT in Theorem FOR.2. SALE can OR be taken DISTRIBUTION to be the entire -plane. Use the famil of solutions in part (a) to find an eplicit solution of the first-order initialvalue problem 2, (0) 0. Even though 0 0 is in the interval ( 2, 2), eplain wh the solution is not defined this interval. & Bartlett Learning, LLC (c) Determine NOT the FOR largest SALE interval OR I of definition DISTRIBUTION for the solution of the initial-value problem in part (b). 3. (a) Verif that /( c) is a one-parameter famil of solutions of the differential equation 2. (b) Since f (, ) 2 and f/ 2 are continuous everwhere, Bartlett the region Learning, R Theorem LLC.2. can be taken to be & 5 & Bartlett Learning, LLC 5 NOT FOR the SALE entire -plane. OR DISTRIBUTION Find a solution from the famil in NOT FIGURE FOR.2.9 SALE Graph OR for DISTRIBUTION FIGURE.2.0 Graph for part (a) that satisfies (0). Find a solution from the Problem 37 Problem 38 famil in part (a) that satisfies (0). Determine the largest interval I of definition for the solution of each In Problems 39 44, c cos 3 c 2 sin 3 is a two-parameter initial-value problem. famil of solutions of the second-order DE 9 0. If possible, LLC find a solution of the differential equation & that Bartlett satisfies Learning, the LLC 32. (a) Find a solution from the famil Jones in part &(a) Bartlett of Problem Learning, 3 that satisfies 2, (0) NOT FOR 0, where SALE 0 0. OR Eplain DISTRIBUTION given side conditions. The conditions NOT specified FOR at SALE two different OR DISTRIBUTI wh the largest interval I of definition for this solution is points are called boundar conditions. either ( q, / 0 ) or (/ 0, q). (b) Determine the largest interval I of definition for the 39. (0) 0, (p/6) 4. (0) 0, (p/4) (0) 0, (p) (0), (p) 5 solution of the first-order initial-value problem 2, 43. (0) 0, (p) 4 & Bartlett 44. (p/3) Learning,, (p) LLC 0 (0) (a) Verif that c is a one-parameter famil of Discussion Problems solutions of the differential equation d/d 3. (b) B hand, sketch the graph of the implicit solution In Problems 45 and 46, use Problem 55 in Eercises. and (2) Find all eplicit solutions f() of the and (3) of this section. & DE Bartlett in part (a) defined Learning, b this LLC relation. Give the interval I NOT FOR of SALE definition OR of each DISTRIBUTION eplicit solution. 45. Jones Find a & function Bartlett Learning, f () whose graph LLC at each point (, ) has NOT the FOR slope SALE given b OR 8e 2 DISTRIBUTION 6 and has the -intercept (0, 9). 8 CHAPTER Introduction to Differential Equations FIGURE.2.7 Graph for Problem FIGURE.2.8 Graph for Problem

19 & Bartlett Learning LLC, an Ascend Learning Compan Find a function f () whose second derivative is 2 2 at each point (, ) on its graph and 5 is & Bartlett tangent to Learning, the graph at the LLC point corresponding to. 6 4, $ Consider the initial-value problem 2, (0) have the same domain but are clearl different. See FIGURES.2.2(a) NOT 2. FOR SALE OR DISTRIBUTION Determine which of the two curves shown in FIGURE.2. is the onl plausible solution curve. Eplain our reasoning. (0, ¹ ₂) and () e 0,, 0 and.2.2(b), respectivel. Show that both functions are solutions of the initial-value problem d/d /2, (2) on the interval ( q, q). Resolve the apparent contradiction between this fact and the last sentence in Eample 5. & Bartlett Learning, LLC & Bartlett Learning, LLC (a) (b) NOT FOR SALE OR DISTRIBUTION FIGURE.2.2 Two solutions of the IVP in Problem 50 FIGURE.2. Graph for Problem Show that 48. Determine a plausible value of 0 for which the graph of the solution of the initial-value problem 2 3 6, ( 0 ) 0 & Bartlett is tangent Learning, to the -ais at LLC ( 0, 0). Eplain our reasoning. & Bartlett Learning, # LLC 0 "t 3 dt NOT FOR 49. SALE Suppose OR that DISTRIBUTION the first-order differential equation d/d f NOT (, ) FOR SALE OR DISTRIBUTION is an implicit solution of the initial-value problem possesses a one-parameter famil of solutions and that f (, ) satisfies the hpotheses of Theorem.2. in some rectangular d 2 region R of the -plane. Eplain wh two different solution 2 d , (0) 0, 9(0). curves cannot intersect or be tangent to each other at a point ( 0, 0 ) in R. Assume that $ 0. [Hint: The integral & Bartlett is nonelementar. Learning, See LLC (ii) in the Remarks at the end of Section..] 50. The functions () 6 4, q,, q (2, ) (2, ).3 Differential Equations as Mathematical Models INTRODUCTION In this section we introduce the notion of a mathematical model. Roughl speaking, a mathematical model is a mathematical description of something. This description could be as simple as a function. For eample, Leonardo da Vinci (452 59) was able to deduce the speed v of a falling bod b a eamining & Bartlett a sequence. Learning, Leonardo allowed LLC water drops to fall, at equall spaced intervals of time, NOT between FOR two SALE boards OR covered DISTRIBUTION with blotting paper. When a spring mechanism was disengaged, the boards were clapped together. See FIGURE.3.. B carefull eamining the sequence of water blots, Leonardo discovered that the distances between consecutive drops increased in a continuous arithmetic proportion. In this manner he discovered the formula v gt. Although there are man kinds of mathematical models, in this section we focus onl on differential equations and discuss some specific differential-equation models & Bartlett in biolog, Learning, phsics, LLC and chemistr. Once we have studied some methods for NOT solving FOR DEs, SALE in Chapters OR 2 DISTRIBUTI and 3 we return to, and solve, some of these models. Mathematical Models It is often desirable to describe the behavior of some real-life sstem or phenomenon, whether phsical, sociological, or even economic, in mathematical terms. & Bartlett Learning, The mathematical LLC description of a sstem or a phenomenon & Bartlett is called Learning, a mathematical LLC model and is constructed with certain goals in mind. NOT For FOR eample, SALE we ma OR wish DISTRIBUTION to understand the mechanisms of a certain ecosstem b studing the growth of animal populations in that sstem, or we ma wish to date fossils b means of analzing the deca of a radioactive substance either in the fossil or in the stratum in which it was discovered. Construction of a mathematical model of a sstem starts with identification of the variables that are responsible for changing the & sstem. Bartlett We ma Learning, choose not to LLC incorporate all these variables into the model at first. In NOT this first FOR step SALE we are specifing OR DISTRIBUTION the level of resolution of the model. Net, FIGURE.3. Da Vinci s apparatus for determining the speed of falling bod.3 Differential Equations as Mathematical Models 9

20 & Bartlett Learning LLC, an Ascend Learning Compan.. we make a set of reasonable assumptions or hpotheses about the sstem we are tring to describe. These assumptions will also include an empirical laws that ma be applicable to the sstem. For some purposes it ma be Jones perfectl & within Bartlett reason to Learning, be content with LLC low-resolution models. For eample, ou ma alread NOT be aware FOR that in SALE modeling OR the DISTRIBUTION motion of a bod falling near the surface of the Earth, the retarding force of air friction, is sometimes ignored in beginning phsics courses; but if ou are a scientist whose job it is to accuratel predict the flight path of a long-range projectile, air resistance and other factors such as the curvature of the Earth have to be taken into account. Since the assumptions made about a sstem frequentl involve a rate of change of one or more of the & variables, Bartlett the Learning, mathematical LLC depiction of all these assumptions ma be & one Bartlett or more equations involving SALE OR derivatives. DISTRIBUTION In other words, the mathematical model NOT ma be FOR a differential SALE equation OR DISTRIBUTIO Learning, LLC NOT FOR or a sstem of differential equations. Once we have formulated a mathematical model that is either a differential equation or a sstem of differential equations, we are faced with the not insignificant problem of tring to solve it. If we can solve it, then we deem the model to be reasonable if its solution is consistent with & Bartlett either Learning, eperimental LLCdata or known facts about the Jones behavior & of Bartlett the sstem. Learning, But if the predictions LLC NOT FOR SALE OR produced DISTRIBUTION b the solution are poor, we can either NOT increase FOR the SALE level of resolution OR DISTRIBUTION of the model or make alternative assumptions about the mechanisms for change in the sstem. The steps of the modeling process are then repeated as shown in FIGURE.3.2. Assumptions and hpotheses Epress assumptions in terms of DEs Mathematical formulation If necessar, alter assumptions or increase resolution of the model Check model predictions with known facts Displa predictions of the model (e.g., graphicall) Solve the DEs Obtain solutions & Bartlett Learning, FIGURE LLC.3.2 Steps in the modeling process Jones & Bartlett Learning, LLC Of course, b increasing the resolution we add to the compleit of the mathematical model and increase the likelihood that we cannot obtain an eplicit solution. A mathematical model of a phsical sstem will often involve the variable time t. A solution of the model then gives the state of the sstem; in other words, for appropriate values of t, the values of the dependent variable (or variables) describe the sstem in the past, present, and future. Population Dnamics NOT One FOR of the SALE earliest attempts OR DISTRIBUTION to model human population growth b means of mathematics was b the English economist Thomas Malthus ( ) in 798. Basicall, the idea of the Malthusian model is the assumption that the rate at which a population of a countr grows at a certain time is proportional* to the total population of the countr at that time. In other words, the more people there are at time t, the more there are going to be in the future. & In Bartlett mathematical Learning, terms, if P(t) LLCdenotes the total population at Jones time t, then & this Bartlett assumption Learning, LLC NOT FOR can be SALE epressed OR as DISTRIBUTION NOT FOR SALE OR DISTRIBUTI 20 CHAPTER Introduction to Differential Equations dp dt r P or dp dt kp, () where k is a constant of proportionalit. This simple model, which fails to take into account man & Bartlett factors Learning, (immigration LLCand emigration, for eample) that can influence & Bartlett human Learning, populations to LLC either NOT FOR SALE OR grow DISTRIBUTION or decline, nevertheless turned out to be NOT fairl accurate FOR SALE in predicting OR DISTRIBUTION the population of the United States during the ears Populations that grow at a rate described b () are rare; nevertheless, () is still used to model growth of small populations over short intervals of time, for eample, bacteria growing in a petri dish. *If two quantities u and v are proportional, we write u ~ v. This means one quantit is a constant multiple of the other: u kv.

21 & Bartlett Learning LLC, an Ascend Learning Compan.. Radioactive Deca The nucleus of an atom consists of combinations of protons and neutrons. Man of these combinations of protons and neutrons are unstable; that is, the atoms deca or transmute into Jones the atoms & Bartlett of another substance. Learning, Such LLC nuclei are said to be radioactive. For eample, over NOT time, the FOR highl SALE radioactive OR radium, DISTRIBUTION Ra-226, transmutes into the radioactive gas radon, Rn-222. In modeling the phenomenon of radioactive deca, it is assumed that the rate da/dt at which the nuclei of a substance decas is proportional to the amount (more precisel, the number of nuclei) A(t) of the substance remaining at time t: da da r A or ka. Jones & Bartlett Learning, (2) LLC dt dt Of course equations () and (2) are eactl the same; the difference is onl in the interpretation of the smbols and the constants of proportionalit. For growth, as we epect in (), k 0, and in the case of (2) and deca, k 0. The model () for growth can be seen as the equation ds/dt rs, which describes the growth of capital S when an annual rate of interest r is Jones compounded & Bartlett continuousl. Learning, The model LLC (2) for deca also occurs in a biological setting, such as determining NOT FOR the SALE half-life OR of a drug the DISTRIBUTION time that it takes for 50% of a drug to be eliminated from a bod b ecretion or metabolism. In chemistr, the deca model (2) appears as the mathematical description of a first-order chemical reaction. The point is this: A single differential equation can serve as a mathematical model for man different phenomena. Mathematical models NOT are FOR often SALE accompanied OR DISTRIBUTION b certain side conditions. For eample, in () and (2) we would epect to know, in turn, an initial population P 0 and an initial amount of radioactive substance A 0 that is on hand. If this initial point in time is taken to be t 0, then we know that P(0) P 0 and A(0) A 0. In other words, a mathematical model can consist of either an initialvalue problem or, as we shall see later in Section 3.9, a boundar-value problem. & Bartlett Newton s Learning, Law of LLC Cooling/Warming According to Newton s empirical law of NOT FOR SALE cooling or OR DISTRIBUTION warming the rate at which the temperature NOT of a bod FOR changes SALE is proportional OR DISTRIBUTIO to the difference between the temperature of the bod and the temperature of the surrounding medium, the so-called ambient temperature. If T(t) represents the temperature of a bod at time t, T m the temperature of the surrounding medium, and dt/dt the rate at which the temperature of the bod changes, then Newton s law of cooling/warming translates into the mathematical statement dt dt r T NOT 2 T dt m FOR or SALE dt k(t OR 2 T m), DISTRIBUTION (3) where k is a constant of proportionalit. In either case, cooling or warming, if T m is a constant, it stands to reason that k 0. Spread of a Disease A & contagious Bartlett disease for Learning, eample, LLCa flu virus is spread throughout a communit b people coming into contact with other people. Let (t) denote the number of people who have contracted the disease and (t) the number of people who have not et been eposed. It seems reasonable to assume that the rate d/dt at which the disease spreads is proportional to the number of encounters or interactions between these two groups of people. If we assume that the number of interactions is jointl proportional to (t) and (t), that is, proportional to the product, then d dt k, NOT FOR SALE OR DISTRIBUTI (4) where k is the usual constant of proportionalit. Suppose a small communit has a fied population of n people. If one infected person is introduced into this communit, then it could be argued that (t) and (t) are related b n. Using this last equation to eliminate (4) gives us the model NOT d k(n FOR SALE 2 ). OR DISTRIBUTION (5) dt An obvious initial condition accompaning equation (5) is (0). Chemical Reactions The disintegration of a radioactive substance, governed b the differential equation (2), Jones is said to & be Bartlett a first-order Learning, reaction. In LLC chemistr, a few reactions follow this same empirical NOT law: If FOR the molecules SALE OR of substance DISTRIBUTION A decompose into smaller molecules, it.3 Differential Equations as Mathematical Models 2

22 & Bartlett Learning LLC, an Ascend Learning Compan.. is a natural assumption that the rate at which this decomposition takes place is proportional to the amount of the first substance that has not undergone conversion; that is, if X(t) is the amount of substance A remaining at an Jones time, then & dx/dt Bartlett kx, Learning, where k is a negative LLC constant since X is decreasing. An eample of a NOT first-order FOR chemical SALE reaction OR DISTRIBUTION is the conversion of t-butl chloride into t-butl alcohol: (CH 3 ) 3 CCl NaOH S (CH 3 ) 3 COH NaCl. Onl the concentration of the t-butl chloride controls the rate of reaction. But in the reaction & Bartlett Learning, CH 3 LLC Cl NaOH S CH 3 OH NaCl, NOT FOR for ever SALE molecule OR of DISTRIBUTION methl chloride, one molecule of sodium NOT hdroide FOR is SALE consumed, OR thus DISTRIBUTIO forming one molecule of methl alcohol and one molecule of sodium chloride. In this case the rate at which the reaction proceeds is proportional to the product of the remaining concentrations of CH 3 Cl and of NaOH. If X denotes the amount of CH 3 OH formed and a and b are the given amounts of the first two chemicals A and B, then the instantaneous amounts not converted to chemical C are a X and b X, respectivel. Hence the rate of formation of C is given b dx k(a 2X)(b 2X), (6) dt where k is a constant of proportionalit. A reaction whose model is equation (6) is said to be second order. NOT FOR input SALE rate of brine OR DISTRIBUTION Mitures The miing NOT of two FOR salt solutions SALE of OR differing DISTRIBUTION concentrations gives rise to a firstorder differential equation for the amount of salt contained in the miture. Let us suppose that a 3 gal/min large miing tank initiall holds 300 gallons of brine (that is, water in which a certain number of pounds of salt has been dissolved). Another brine solution is pumped into the large tank at a constant rate of 3 gallons per minute; the concentration of the salt in this inflow is 2 pounds of salt per 300 gal gallon. & When Bartlett the solution Learning, the tank LLC is well stirred, it is pumped out at the same & rate Bartlett as the entering solution. SALE See OR FIGURE DISTRIBUTION.3.3. If A(t) denotes the amount of salt (measured NOT FOR in pounds) SALE in the OR tank DISTRIBUTIO Learning, LLC NOT FOR at time t, then the rate at which A(t) changes is a net rate: 22 CHAPTER Introduction to Differential Equations da dt rate output rate ainput b 2 a b R of salt of salt in 2 R out. (7) concentration of salt input rate input rate in inflow of brine of salt T T T R in Jones (2 lb/gal) & Bartlett (3 gal/min) Learning, (6 lb/min). LLC output rate of brine 3 gal/min & Bartlett The Learning, input rate RLLC FIGURE.3.3 Miing tank in at which the salt enters the tank is the product of the inflow concentration of NOT FOR SALE OR salt DISTRIBUTION and the inflow rate of fluid. Note that R in is measured in pounds per minute: Now, since the solution is being pumped out of the tank at the same rate that it is pumped in, the number of gallons of brine in the tank at time t is a constant 300 gallons. Hence the concentration of the salt in the tank, as well as in the outflow, is c(t) A(t)/300 lb/gal, and so the output rate R out of salt is concentration of salt output rate output rate & Bartlett Learning, LLC in outflow of brine of NOT salt FOR SALE OR DISTRIBUTI T T T R out a A(t) A(t) lb/galb (3 gal/min) lb/min. & Bartlett The Learning, net rate (7) LLC then becomes da dt 6 2 A NOT da FOR or 00 dt SALE OR DISTRIBUTION A 6. (8) 00 If r in and r out denote general input and output rates of the brine solutions,* respectivel, then there are three possibilities: r in r out, r in r out, and r in r out. In the analsis leading to (8) we *Don t confuse these smbols NOT with R in FOR and R out SALE, which are OR input DISTRIBUTION and output rates of salt.

23 & Bartlett Learning LLC, an Ascend Learning Compan.. have assumed that r in r out. In the latter two cases, the number of gallons of brine in the tank is either increasing (r in r out ) or decreasing (r in r out ) at the net rate r in r out. See Problems 0 2 in Eercises.3. Draining a Tank In hdrodnamics, Torricelli s law states that the speed v of efflu of water through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the A w speed that a bod (in this case a drop of water) would acquire in falling freel from a height h; that is, v!2gh, where g is the acceleration due to gravit. This last epression comes from & Bartlett equating the Learning, kinetic energ LLC 2 mv 2 with the potential energ mgh Jones and solving & Bartlett for v. Suppose Learning, a tank LLC h filled with water is allowed to drain through a hole under the influence of gravit. We would like A h to find the depth h of water remaining in the tank at time t. Consider the tank shown in FIGURE.3.4. If the area of the hole is A h (in ft 2 ) and the speed of the water leaving the tank is v!2gh (in ft/s), then the volume of water leaving the tank per second is A h!2gh (in ft 3 /s). Thus if V(t) denotes the volume of water in the tank at time t, FIGURE.3.4 Jones Water & draining Bartlett from a Learning, tank LLC dv NOT dt A FOR h!2gh, SALE OR DISTRIBUTION (9) where the minus sign indicates that V is decreasing. Note here that we are ignoring the possibilit of friction at the hole that might cause a reduction of the rate of flow there. Now if the tank is such that the volume of water in it at time t can be written V(t) A w h, where A w (in ft 2 ) is the constant area of the upper Jones surface & of Bartlett the water (see Learning, Figure.3.4), LLC then dv/dt A w dh/dt. Substituting this last epression NOT into (9) FOR gives SALE us the desired OR DISTRIBUTION differential equation for the height of the water at time t: dh dt A h!2gh. (0) A w L E(t) R It is interesting to note that (0) remains valid even when A w is not constant. In this case we must NOT FOR SALE epress OR the DISTRIBUTION upper surface area of the water as a function NOT of h; that FOR is, ASALE w A(h). OR See Problem DISTRIBUTIO 4 in Eercises.3. C (a) LRC-series circuit Series Circuits Consider the single-loop LRC-series circuit containing an inductor, resistor, and capacitor shown in FIGURE.3.5(a). The current & in Bartlett a circuit after Learning, a switch is LLC closed is denoted Inductor b i(t); the charge on a capacitor at time t is denoted b q(t). The letters L, R, and C are known inductance L: henrs (h) as inductance, resistance, and capacitance, NOT FOR respectivel, SALE and OR are DISTRIBUTION generall constants. Now according to Kirchhoff s second law, the impressed voltage E(t) on a closed loop must equal the voltage drop across: L di dt sum of the voltage drops in the loop. Figure.3.5(b) also shows the smbols and the formulas for the respective voltage drops across an inductor, a resistor, and a capacitor. Since current i(t) i is related to charge q(t) on the capacitor b i dq/dt, b adding the three voltage drops & Bartlett Learning, L LLC Inductor Resistor Capacitor NOT FOR L di SALE dt L d OR DISTRIBUTION 2 q dt, dq Resistor ir R 2 dt, C q resistance R: ohms (Ω) voltage drop across: ir and equating the sum to the impressed voltage, we obtain a second-order differential equation L d 2 q dq R dt 2 dt i q E(t). () R C NOT FOR SALE OR DISTRIBUTI We will eamine a differential equation analogous to () in great detail in Section 3.8. Capacitor capacitance C: farads (f) Falling Bodies In constructing a mathematical model of the motion of a bod moving voltage drop across: q & Bartlett C Learning, a force LLCfield, one often starts with Newton s Jones second & Bartlett law of motion. Learning, Recall LLC from elementar phsics that Newton s first law of motion NOT FOR states that SALE a bod OR will DISTRIBUTION either remain at rest or will i continue to move with a constant velocit unless acted upon b an eternal force. In each case C this is equivalent to saing that when the sum of the forces F F k that is, the net or resultant force acting on the bod is zero, then the acceleration a of the bod is zero. Newton s (b) Smbols and voltage drops FIGURE.3.5 Current i(t) and charge q(t) second law of motion indicates that when the net force acting on a bod is not zero, then the & are Bartlett measured in Learning, amperes (A) and LLC net force is proportional to its acceleration & Bartlett a, Learning, or more precisel, LLCF ma, where m is the mass NOT FOR coulombs SALE (C), OR respectivel DISTRIBUTION of the bod..3 Differential Equations as Mathematical Models 23

24 & Bartlett Learning LLC, an Ascend Learning Compan.. v Now suppose a rock is tossed upward from a roof of a building as illustrated in FIGURE What is the position s(t) of the rock relative to the ground at time t? The acceleration of the rock is the second derivative d 2 s/dt 2 Jones. If we assume & Bartlett that the Learning, upward direction LLC is positive and that no rock force acts on the rock other NOT than the FOR force of SALE gravit, OR then DISTRIBUTION Newton s second law gives m d 2 s dt mg or d 2 s g. (2) 2 2 dt s 0 s(t) In other words, the net force is simpl the weight F F W of the rock near the surface of the Earth. & Bartlett Recall that Learning, the magnitude LLC of the weight is W mg, where m is the & mass Bartlett of the bod Learning, LLC building NOT FOR and g is SALE the acceleration OR DISTRIBUTION due to gravit. The minus sign in (2) is NOT used because FOR the SALE weight OR of the DISTRIBUTIO ground rock is a force directed downward, which is opposite to the positive direction. If the height of the FIGURE.3.6 Position of rock measured building is s 0 and the initial velocit of the rock is v 0, then s is determined from the second-order from ground level initial-value problem d 2 s dt g, s(0) Jones s 0, s9(0) & Bartlett v 2 0. Learning, LLC(3) kv Although we have not stressed solutions of the equations we have constructed, we note that (3) positive direction can be solved b integrating the constant g twice with respect to t. The initial conditions determine the two constants of integration. You might recognize the solution of (3) from elemen- air resistance tar phsics as the formula s(t) 2 gt 2 v 0 t s 0. Falling Bodies and NOT Air Resistance FOR SALE Prior OR DISTRIBUTION to the famous eperiment b Italian gravit mathematician and phsicist Galileo Galilei ( ) from the Leaning Tower of Pisa, it was generall believed that heavier objects in free fall, such as a cannonball, fell with a greater mg acceleration than lighter objects, such as a feather. Obviousl a cannonball and a feather, when FIGURE.3.7 Falling bod of mass m dropped simultaneousl from the same height, do fall at different rates, but it is not because a cannonball & Bartlett is heavier. Learning, The difference LLCin rates is due to air resistance. The resistive & Bartlett force of Learning, air LLC NOT FOR was ignored SALE in OR the model DISTRIBUTION given in (3). Under some circumstances NOT FOR a falling SALE bod of OR mass DISTRIBUTIO m such as a feather with low densit and irregular shape encounters air resistance proportional to its instantaneous velocit v. If we take, in this circumstance, the positive direction to be oriented downward, then the net force acting on the mass is given b F F F 2 mg kv, where the weight F mg of the bod is a force acting in the positive direction and air resistance Learning, F 2 kv LLC is a force, called viscous damping, or drag, & Bartlett acting in the Learning, opposite or LLC upward & Bartlett NOT FOR SALE OR direction. DISTRIBUTION See FIGURE.3.7. Now since v is related NOT to acceleration FOR SALE a b a OR dv/dt, DISTRIBUTION Newton s second law becomes F ma m dv/dt. B equating the net force to this form of Newton s second law, (a) Telephone wires we obtain a first-order differential equation for the velocit v(t) of the bod at time t, m dv mg 2 kv. (4) & dtbartlett Learning, LLC Here k is a positive constant NOT of proportionalit FOR SALE called OR the DISTRIBUTION drag coefficient. If s(t) is the distance the bod falls in time t from its initial point of release, then v ds/dt and a dv/dt d 2 s/dt 2. In terms of s, (4) is a second-order differential equation (b) Suspension bridge m d 2 s ds mg 2 k or m d 2 s ds FIGURE.3.8 Cables suspended between k mg. (5) 2 2 & Bartlett Learning, dt LLCdt dt dtjones & Bartlett Learning, LLC vertical supports NOT FOR SALE OR DISTRIBUTI Suspended Cables Suppose a fleible cable, wire, or heav rope is suspended between two vertical supports. Phsical eamples of this could be a long telephone wire strung between two posts as shown in red in FIGURE.3.8(a), or one of the two cables supporting the roadbed of a T 2 T2 suspension bridge shown in red in Figure.3.8(b). Our goal is to construct a mathematical model sin θ & Bartlett that Learning, describes the LLC shape that such a cable assumes. P 2 θ To begin, let s agree to eamine onl a portion or element of the cable between its lowest point wire NOT FOR SALE T 2 cos θor DISTRIBUTION P and an arbitrar point P 2. As drawn in blue in FIGURE.3.9, this element of the cable is the P T curve in a rectangular coordinate sstem with the -ais chosen to pass through the lowest point (0, a) P on the curve and the -ais chosen a units below P. Three forces are acting on the cable: the W tensions T and T 2 in the cable that are tangent to the cable at P and P 2, respectivel, and the (, 0) portion W of the total vertical load & between Bartlett the Learning, points P and LLCP 2. Let T T, NOT FIGURE FOR.3.9 SALE Element OR of cable DISTRIBUTION T 2 T 2, and W W denote NOT the FOR magnitudes SALE of OR these DISTRIBUTION vectors. Now the tension T 2 resolves 24 CHAPTER Introduction to Differential Equations

25 & Bartlett Learning LLC, an Ascend Learning Compan.. into horizontal and vertical components (scalar quantities) T 2 cos u and T 2 sin u. Because of static equilibrium, we can write T& Bartlett T 2 cos u and Learning, W T LLC 2 sin u. B dividing the last equation b the first, we eliminate T 2 and get tan u W/T. But since d/d tan u, we arrive at d d W. (6) T This simple first-order differential equation serves as a model for both the shape of a fleible wire such as a telephone wire hanging under its own weight as well as the shape of the cables that support the roadbed. We will come back to equation (6) in Eercises 2.2 and in Section 3.. & Bartlett Learning, REMARKS LLC Each eample in this section has described NOT a dnamical FOR SALE sstem: OR a sstem DISTRIBUTION that changes or evolves with the flow of time t. Since the stud of dnamical sstems is a branch of mathematics currentl in vogue, we shall occasionall relate the terminolog of that field to the discussion at hand. In more precise terms, a dnamical sstem consists of a set of time-dependent variables, called state variables, together with a rule that enables us to determine (without ambiguit) the state of the sstem (this ma be past, present, or future states) in terms of a state prescribed at some time t 0. Dnamical NOT FOR sstems SALE are classified OR as DISTRIBUTION either discrete-time sstems or continuous-time sstems. In this course we shall be concerned onl with continuous-time dnamical sstems sstems in which all variables are defined over a continuous range of time. The rule or the mathematical model in a continuous-time dnamical sstem is a differential equation or a sstem of differential equations. The state of the sstem at a time t is the value of the state variables at that time; the specified state of the sstem at a time t 0 is simpl the initial conditions that accompan OR the DISTRIBUTION mathematical model. The solution of the initial-value NOT FOR problem SALE is referred OR DISTRIBUTIO to as the NOT FOR SALE response of the sstem. For eample, in the preceding case of radioactive deca, the rule is da/dt ka. Now if the quantit of a radioactive substance at some time t 0 is known, sa A(t 0 ) A 0, then b solving the rule, the response of the sstem for t t 0 is found to be A(t) A 0 e t2t 0 (see Section 2.7). The response A(t) is the single-state variable for this sstem. In the case of the rock tossed from the roof of the building, the response of the sstem, the solution of the differential equation d 2 s/dt NOT 2 g FOR subject SALE to the initial OR DISTRIBUTION state s(0) s 0, s (0) v 0, is the function s(t) 2gt 2 v 0 t s 0, 0 t T, where the smbol T represents the time when the rock hits the ground. The state variables are s(t) and s (t), which are, respectivel, the vertical position of the rock above ground and its velocit at time t. The acceleration s (t) is not a state variable since we onl have to know an initial position and initial velocit at a time t 0 to uniquel determine the rock s position Jones s(t) & and Bartlett velocit Learning, s (t) v(t) for LLC an time in the interval [t 0, T ]. The acceleration s (t) NOT a(t) is, FOR of course, SALE given OR b the DISTRIBUTION differential equation s (t) g, 0 t T. One last point: Not ever sstem studied in this tet is a dnamical sstem. We shall also eamine some static sstems in which the model is a differential equation..3 Eercises NOT FOR Answers SALE to selected OR DISTRIBUTION odd-numbered problems begin on page NOT ANS-. FOR SALE OR DISTRIBUTI Population Dnamics that the rate at which the population changes is a net rate that is, the difference between the rate of births and the rate of. Under the same assumptions underling the model in (), determine a differential & Bartlett equation Learning, governing the LLC growing popula- deaths in the communit. Determine a model for the population P(t) Jones if both & the Bartlett birth rate and Learning, the death rate LLC are proportional tion P(t) of a countr when individuals are allowed to immigrate to NOT the population FOR SALE present OR at time DISTRIBUTION t. into the countr at a constant rate r 0. What is the differential equation for the population P(t) of the countr when indi- 3. Using the concept of a net rate introduced in Problem 2, determine a differential equation governing a population P(t) if viduals are allowed to emigrate at a constant rate r 0? the birth rate is proportional to the population present at time 2. The population model given in () fails to take death into t but the death rate is proportional to the square of the population present Learning, at time t. LLC & Bartlett consideration; Learning, the growth LLC rate equals the birth rate. In another & Bartlett NOT FOR SALE model OR of a changing DISTRIBUTION population of a communit, it is assumed.3 Differential Equations as Mathematical Models 25

26 & Bartlett Learning LLC, an Ascend Learning Compan.. 4. Modif the model in Problem 3 for the net rate at which the Mitures population P(t) of a certain kind of fish changes b also assuming & Bartlett that the fish Learning, are harvested LLC at a constant rate h Suppose that a large miing tank initiall holds 300 gallons of water in which 50 pounds of salt has been dissolved. Pure NOT Newton s FOR SALE Law of OR Cooling/Warming DISTRIBUTION NOT water FOR is SALE pumped OR into the DISTRIBUTION tank at a rate of 3 gal/min, and when the solution is well stirred, it is pumped out at the same rate. 5. A cup of coffee cools according to Newton s law of cooling Determine a differential equation for the amount A(t) of salt (3). Use data from the graph of the temperature T(t) in in the tank at time t. What is A(0)? FIGURE.3.0 to estimate the constants T m, T 0, and k in a model of the form of the first-order initial-value Jones & problem Bartlett Learning, 0. LLC Suppose that a large miing tank initiall & holds Bartlett 300 gallons Learning, LLC of water in which 50 pounds of salt has been dissolved. dt dt k(t 2 T Another brine solution is pumped into the tank at a rate of 3 m), T(0) T 0. gal/min, and when the solution is well stirred, it is pumped T out at a slower rate of 2 gal/min. If the concentration of the solution entering is 2 lb/gal, determine a differential equation 200 for the amount A(t) of salt & in Bartlett the tank at Learning, time t. LLC 50. What is the NOT differential FOR equation SALE in OR Problem DISTRIBUTION 0 if the wellstirred solution is pumped out at a faster rate of 3.5 gal/min? Generalize the model given in (8) of this section b assuming that the large tank initiall contains N 50 0 number of gallons of brine, r in and r out are the input and output rates of the brine, respectivel (measured & Bartlett in gallons Learning, per minute), c in LLC is the concentration of the t NOT FOR SALE OR salt in the inflow, c(t) is the concentration of the salt in the tank min DISTRIBUTION as well as in the outflow at time t (measured in pounds of salt FIGURE.3.0 Cooling curve in Problem 5 per gallon), and A(t) is the amount of salt in the tank at time t. 6. The ambient temperature T m in (3) could be a function of time t. Suppose that in an artificiall controlled environment, Draining a Tank T m (t) is periodic with a 24-hour period, & as Bartlett illustrated Learning, in LLC 3. Suppose water is leaking from a tank through a circular hole of FIGURE.3.. Devise a mathematical NOT FOR model for SALE the temperature OR DISTRIBUTION area A h at its bottom. When water NOT leaks FOR through SALE a hole, OR friction DISTRIBUTIO T(t) of a bod within this environment. and contraction of the stream near the hole reduce the volume T m (t) of the water leaving the tank per second to ca h "2gh, where c (0 c ) is an empirical constant. Determine a differential equation for the Jones height h & of Bartlett water at time Learning, t for the cubical LLC tank & Bartlett Learning, 48 t LLC Midnight Noon Midnight Noon Midnight FIGURE.3. Ambient temperature in Problem 6 4. The right-circular conical tank shown in FIGURE.3.3 loses Spread of a Disease/Technolog water out of a circular hole at its bottom. Determine a differential equation for the height of the water h at time t. The 7. Suppose a student carring a flu Jones virus returns & Bartlett to an isolated Learning, LLC radius of the hole is 2 in, g 32 ft/s Jones college campus of 000 students. Determine a differential 2, and the & friction/contraction factor introduced in Problem NOT 3 FOR is c 0.6. SALE OR DISTRIBUTI Bartlett Learning, LLC equation governing the number of students (t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between the number 8 ft of students with the flu and the number of students who have A w not et been eposed to & it. Bartlett Learning, LLC 8. At a time t 0, a technological innovation is introduced into 20 ft NOT hfor SALE OR DISTRIBUTION a communit with a fied population of n people. Determine a differential equation governing the number of people (t) who have adopted the innovation at time t if it is assumed that the rate at which the innovation spreads through the communit is & jointl Bartlett proportional Learning, to the number LLC of people who have circular hole NOT FOR adopted SALE it and the OR number DISTRIBUTION of people who have not adopted it. NOT FIGURE FOR.3.3 SALE Conical OR tank DISTRIBUTION in Problem 4 26 CHAPTER Introduction to Differential Equations in FIGURE.3.2. The radius of the hole is 2 in and g 32 ft/s 2. h circular hole A w 0 ft FIGURE.3.2 Cubical tank in Problem 3

27 & Bartlett Learning LLC, an Ascend Learning Compan.. Series Circuits 5. A series circuit contains a resistor and an inductor as shown & Bartlett in FIGURE.3.4. Learning, Determine LLC a differential equation for the current i(t) OR if the DISTRIBUTION resistance is R, the inductance is L, and NOT the FOR SALE OR DISTRIBUTION 0 surface NOT FOR SALE impressed voltage is E(t). s/2 s/2 0 (t) E L R FIGURE.3.4 LR-series circuit in Problem 5 6. A series circuit contains a resistor and a capacitor as shown in FIGURE Jones.3.5. & Bartlett Determine Learning, a differential LLC equation for the charge NOT q(t) FOR on the SALE capacitor OR if DISTRIBUTION the resistance is R, the capacitance is C, and the impressed voltage is E(t). E R C FIGURE.3.5 RC-series circuit in Problem 6 (a) FIGURE.3.7 Bobbing motion of floating barrel in Problem 8 Newton s Second Law and Hooke s Law 9. After a mass m is attached to a spring, it stretches s units and then hangs at rest in the equilibrium position as shown in FIGURE.3.8(b). After the spring/mass sstem has been set in motion, NOT let FOR (t) denote SALE the directed OR DISTRIBUTION distance of the mass beond the equilibrium position. As indicated in Figure.3.8(c), assume that the downward direction is positive, that the motion takes place in a vertical straight line through the center of & Bartlett gravit of the Learning, mass, and LLC that the onl forces acting on the sstem are the weight of the mass and the restoring force of NOT FOR SALE the stretched OR DISTRIBUTION spring. Use Hooke s law: The restoring force of a spring is proportional to its total elongation. Determine a differential equation for the displacement (t) at time t. (b) Falling Bodies and Air Resistance & Bartlett Learning, LLC 7. For high-speed motion NOT through FOR the SALE air such OR as DISTRIBUTION the skdiver shown in FIGURE.3.6 falling before the parachute is opened air resistance is closer to a power of the instantaneous velocit v(t). Determine a differential equation for the velocit v(t) of a falling bod of mass m if air resistance is proportional to the square Jones of & the Bartlett instantaneous Learning, velocit. LLC kv 2 unstretched spring (a) s (t) < 0 m = 0 (t) > 0 equilibrium position m (b) (c) FIGURE.3.8 Spring/mass sstem in Problem 9 mg 20. In Problem 9, what is a differential equation for the displacement (t) if the motion takes place in a medium that imparts a damping force on the spring/mass sstem that is proportional to the instantaneous velocit of the mass and acts in a direction FIGURE.3.6 Air resistance proportional to square opposite to that of motion? of velocit in Problem 7 Newton s Second Law NOT and FOR Archimedes SALE OR Principle DISTRIBUTION Newton s Second Law NOT and FOR Variable SALE Mass OR DISTRIBUTI When the mass m of a bod moving through a force field is variable, 8. A clindrical barrel s ft in diameter of weight w lb is floating Newton s second law of motion takes on the form: If the net force in water as shown in FIGURE.3.7(a). After an initial depression, the barrel ehibits an up-and-down bobbing motion along acting on a bod is not zero, then the net force F is equal to the time rate of change of momentum of the bod. That is, a vertical Jones line. & Using Bartlett Figure.3.7(b), Learning, determine LLCa differential equation NOT FOR for the SALE vertical displacement OR DISTRIBUTION (t) if the origin is taken NOT FOR SALE FOR d DISTRIBUTION to be on the vertical ais at the surface of the water when the dt (mv)*, (7) barrel is at rest. Use Archimedes principle: Buoanc, or upward force of the water on the barrel, is equal to the weight where mv is momentum. Use this formulation of Newton s second of the water displaced. Assume that the downward direction law in Problems 2 and 22. & Bartlett is positive, Learning, that the weight LLC densit of water is 62.4 lb/ft 3, and Jones & Bartlett Learning, LLC NOT FOR SALE that there OR is DISTRIBUTION no resistance between the barrel and the water. NOT FOR *Note SALE that when OR m DISTRIBUTION is constant, this is the same as F ma..3 Differential Equations as Mathematical Models 27

28 & Bartlett Learning LLC, an Ascend Learning Compan.. 2. Consider a single-stage rocket that is launched verticall upward as shown in the accompaning photo. Let m(t) denote the total & Bartlett mass of the Learning, rocket at time LLC t (which is the sum of three NOT FOR masses: SALE the constant OR DISTRIBUTION mass of the paload, the constant mass of the vehicle, and the variable amount of fuel). Assume that the positive direction is upward, air resistance is proportional to the instantaneous velocit v of the rocket, and R is the upward thrust or force generated b the propulsion sstem. Use (7) to find a mathematical model for the velocit v(t) of the rocket. 24. Suppose a hole is drilled through the center of the Earth and a bowling ball of mass m is dropped into the hole, as shown in FIGURE & Bartlett Construct Learning, a mathematical LLCmodel that describes NOT the FOR motion SALE of the OR ball. DISTRIBUTION At time t let r denote the distance from the center of the Earth to the mass m, M denote the mass of the Earth, M r denote the mass of that portion of the Earth within a sphere of radius r, and d denote the constant densit of the Earth. R r surface m FIGURE.3.20 NOT Hole through FOR SALE Earth in Problem OR DISTRIBUTION 24 Additional Mathematical Models 25. Learning Theor In the theor of learning, the rate at which a Sebastian Kaulitzki/ShutterStock, Inc. subject is memorized is assumed to be proportional to the amount Rocket & in Bartlett Problem 2Learning, LLC that is & left Bartlett to be memorized. Learning, Suppose LLC M denotes the total amount NOT 22. FOR In Problem SALE 2, OR suppose DISTRIBUTION m(t) m p m v m f (t) where m p is NOT of FOR a subject SALE to be memorized OR DISTRIBUTION and A(t) is the amount memorized constant mass of the paload, m v is the constant mass of the in time t. Determine a differential equation for the amount A(t). vehicle, and m f (t) is the variable amount of fuel. 26. Forgetfulness In Problem 25, assume that the rate at which (a) Show that the rate at which the total mass of the rocket material is forgotten is proportional to the amount memorized changes is the same as the rate at which the mass of the in time t. Determine a differential equation for A(t) when forgetfulness is taken into account. fuel changes. (b) If the rocket consumes NOT its fuel FOR at a constant SALE rate OR l, find DISTRIBUTION 27. Infusion of a Drug A drug is NOT infused FOR into a SALE patient s OR bloodstream at a constant rate of r grams per second. Simultaneousl, DISTRIBUTIO m(t). Then rewrite the differential equation in Problem 2 in terms of l and the initial total mass m(0) m 0. the drug is removed at a rate proportional to the amount (t) (c) Under the assumption in part (b), show that the burnout of the drug present at time t. Determine a differential equation time t b 0 of the rocket, or the time at which all the fuel governing the amount (t). is consumed, is t b & Bartlett m f (0)/l, where Learning, m f (0) is the LLC initial mass of the fuel. 28. Tractri A motorboat starts at the origin and moves in the direction of NOT the positive FOR -ais, SALE pulling OR DISTRIBUTION a waterskier along a Newton s Second Law and the Law of Universal Gravitation curve C called a tractri. See FIGURE.3.2. The waterskier, initiall located on the -ais at the point (0, s), is pulled b keeping a rope of constant length s, which is kept taut throughout the motion. At time t. 0 the waterskier is at the point 23. B Newton s law of universal gravitation, the free-fall acceleration & a Bartlett of a bod, such Learning, as the satellite LLCshown in FIGURE.3.9, P(, ). Find the differential equation of the path of motion C. NOT FOR falling SALE a great distance OR DISTRIBUTION to the surface is not the constant g. Rather, NOT FOR SALE OR DISTRIBUTION the acceleration a is inversel proportional to the square of the distance from the center of the Earth, a k/r 2 (0, s), where k is the constant of proportionalit. Use the fact that at the surface of P(, ) the Earth r R and a g to determine k. If the positive direction is upward, use Newton s second Jones law and & Bartlett his universal Learning, law LLC waterskier of gravitation to find a differential NOT equation FOR SALE for the distance OR DISTRIBUTION r. s NOT FOR SALE OR DISTRIBUTI satellite of C θ mass m motorboat surface Earth of mass M r R NOT FOR FIGURE SALE.3.9 Satellite OR DISTRIBUTION in Problem 23 FIGURE.3.2 Tractri curve in Problem Reflecting NOT Surface FOR Assume SALE that OR when DISTRIBUTION the plane curve C shown in FIGURE.3.22 is revolved about the -ais it generates a surface of revolution with the propert that all light ras L parallel to the -ais striking the surface are reflected to a single point O (the origin). Use the fact that the angle of incidence is equal to the angle of reflection to determine a differential equation that describes the shape of the curve C. Such a curve C is 28 CHAPTER Introduction to Differential Equations

29 & Bartlett Learning LLC, an Ascend Learning Compan.. important in applications ranging from construction of telescopes w to satellite antennas, automobile headlights, and solar collectors. curve C of intersection of -plane and & Bartlett [Hint: Inspection Learning, of the figure LLCshows that we can write f 2u. Jones & Bartlett Learning, surface of revolution LLC NOT FOR SALE Wh? OR Now use DISTRIBUTION an appropriate trigonometric identit.] P tangent P(, ) θ θ C L & Bartlett Learning, LLC 35. Falling Bod In Problem 23, suppose r R s, where s is φ the distance from the surface of the Earth to the falling bod. O What does the differential equation obtained in Problem 23 FIGURE.3.22 Reflecting & Bartlett surface Learning, in Problem 29LLC become when s & is Bartlett ver small Learning, compared to R? LLC 36. Raindrops NOT FOR Keep Falling SALE In OR meteorolog, DISTRIBUTION the term virga refers Discussion Problems to falling raindrops or ice particles that evaporate before the reach the ground. Assume that a tpical raindrop is spherical 30. Reread Problem 45 in Eercises. and then give an eplicit in shape. Starting at some time, which we can designate as solution P(t) for equation (). Find a one-parameter famil of t 0, the raindrop of radius r solutions of (). 0 falls from rest from a cloud & Bartlett and begins Learning, to evaporate. LLC NOT FOR 3. SALE Reread OR the sentence DISTRIBUTION following equation (3) and assume NOT that FOR SALE (a) If it OR is assumed DISTRIBUTION that a raindrop evaporates in such a manner T m is a positive constant. Discuss wh we would epect that its shape remains spherical, then it also makes sense to k 0 in (3) in both cases of cooling and warming. You might assume that the rate at which the raindrop evaporates that start b interpreting, sa, T(t) T m in a graphical manner. is, the rate at which it loses mass is proportional to its 32. Reread the discussion leading up to equation (8). If we assume surface area. Show that this latter assumption implies that that initiall the tank holds, Jones sa, 50 & lbs Bartlett of salt, it stands Learning, to reason LLC the rate at which the Jones radius r & of Bartlett the raindrop Learning, decreases is LLC that since salt is being NOT added FOR to the SALE tank continuousl OR DISTRIBUTION for t 0, a constant. Find r(t). NOT [Hint: FOR See Problem SALE 55 OR in Eercises DISTRIBUTIO..] that A(t) should be an increasing function. Discuss how ou (b) If the positive direction is downward, construct a mathematical model for the velocit v of the falling raindrop might determine from the DE, without actuall solving it, the number of pounds of salt in the tank after a long period of time. at time t. Ignore air resistance. [Hint: Use the form of 33. Population Model The differential equation dp/dt (k cos t)p, Newton s second law as given in (7).] where k is a positive & Bartlett constant, Learning, is a model of LLC human population 37. Let It Snow Jones The & snowplow Bartlett problem Learning, is a classic LLCand appears P(t) NOT of a FOR certain SALE communit. OR Discuss DISTRIBUTION an interpretation for the in NOT man differential FOR SALE equations OR DISTRIBUTION tets but was probabl made solution of this equation; in other words, what kind of population do ou think the differential equation describes? One da it started snowing at a heav and stead rate. A famous b Ralph Palmer Agnew: 34. Rotating Fluid As shown in FIGURE.3.23(a), a right-circular snowplow started out at noon, going 2 miles the first hour clinder partiall filled with fluid is rotated with a constant and mile the second hour. What time did it start snowing? angular velocit v about a vertical -ais through its center. The NOT FOR SALE rotating OR fluid DISTRIBUTION is a surface of revolution S. To identif S we NOT first FOR SALE If possible, OR find DISTRIBUTION the tet Differential Equations, Ralph Palmer Agnew, McGraw-Hill, and then discuss the construction and establish a coordinate sstem consisting of a vertical plane determined b the -ais and an -ais drawn perpendicular to the solution of the mathematical model. -ais such that the point of intersection of the aes (the origin) is located at the lowest point on the surface S. We then seek a function f (), which represents the curve C of intersection of the surface S and NOT the vertical FOR coordinate SALE plane. OR DISTRIBUTION Let the point NOT FOR SALE OR DISTRIBUTI P(, ) denote the position of a particle of the rotating fluid of mass m in the coordinate plane. See Figure.3.23(b). (a) At P, there is a reaction force of magnitude F due to the other particles of the fluid, which is normal to the surface S. B Newton s second law the magnitude of the net force NOT acting FOR on the SALE particle OR is mvdistribution 2. What is this force? Use Figure.3.23(b) to discuss the nature and origin of the equations F cos u mg, F sin u mv 2. FIGURE.3.23 Rotating fluid in Problem 34 & Bartlett (b) Use part Learning, (a) to find a LLC first-order differential equation that Jones & Bartlett aetb/istock/thinkstock Learning, LLC NOT FOR SALE defines OR DISTRIBUTION the function f (). NOT FOR SALE Snowplow OR in Problem DISTRIBUTION 37 (a) tangent line to curve C at P.3 Differential Equations as Mathematical Models 29 (b) θ mw 2 P(, ) θ mg