INTRODUCTION TO DIFFERENTIAL EQUATIONS

INTRODUCTION TO DIFFERENTIAL EQUATIONS. Definitions and Terminolog. Initial-Value Problems.3 Differential Equations as Mathematical Models CHAPTER IN REVIEW The words differential and equations certainl suggest solving some kind of equation that contains derivatives,,.... Analogous to a course in algebra and trigonometr, in which a good amount of time is spent solving equations such as 5 4 0 for the unknown number, in this course one of our tasks will be to solve differential equations such as 0 for an unknown function (). The preceding paragraph tells something, but not the complete stor, about the course ou are about to begin. As the course unfolds, ou will see that there is more to the stu of differential equations than just mastering methods that someone has devised to solve them. But first things first. In order to read, stu, and be conversant in a specialized subject, ou have to learn the terminolog of that discipline. This is the thrust of the first two sections of this chapter. In the last section we briefl eamine the link between differential equations and the real world. Practical questions such as How fast does a disease spread? How fast does a population change? involve rates of change or derivatives. As so the mathematical description or mathematical model of eperiments, observations, or theories ma be a differential equation.

CHAPTER INTRODUCTION TO DIFFERENTIAL EQUATIONS. DEFINITIONS AND TERMINOLOGY REVIEW MATERIAL Definition of the derivative Rules of differentiation Derivative as a rate of change First derivative and increasing/decreasing Second derivative and concavit INTRODUCTION The derivative d of a function () is itself another function () found b an appropriate rule. The function e 0. is differentiable on the interval (, ), and b the Chain Rule its derivative is >d 0.e 0.. If we replace e 0. on the right-hand side of the last equation b the smbol, the derivative becomes 0.. () d Now imagine that a friend of ours simpl hands ou equation () ou have no idea how it was constructed and asks, What is the function represented b the smbol? You are now face to face with one of the basic problems in this course: How do ou solve such an equation for the unknown function ()? A DEFINITION The equation that we made up in () is called a differential equation. Before proceeding an further, let us consider a more precise definition of this concept. DEFINITION.. 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). To talk about them, we shall classif differential equations b tpe, order, and linearit. CLASSIFICATION BY TYPE If an equation contains onl ordinar derivatives of one or more dependent variables with respect to a single independent variable it is said to be an ordinar differential equation (ODE). For eample, d 5 e, A DE can contain more than one dependent variable d d d 6 0, and d dt dt b b () are ordinar differential equations. An equation involving partial derivatives of one or more dependent variables of two or more independent variables is called a

. DEFINITIONS AND TERMINOLOGY 3 partial differential equation (PDE). For eample, u u 0, u u u t t, and u v (3) are partial differential equations. * Throughout this tet ordinar derivatives will be written b using either the Leibniz notation d, d d, d 3 d 3,... or theprime notation,,,... B using the latter notation, the first two differential equations in () can be written a little more compactl as 5 e and 6 0. 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 of is written 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 equation unknown function or dependent variable d 6 0 dt independent variable it is immediatel seen that the smbol now represents a dependent variable, whereas the independent variable is t. You should also be aware that in phsical sciences and engineering, Newton s 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 s dt 3 becomes s 3. Partial derivatives are often denoted b a subscript notation indicating the independent variables. For eample, with the subscript notation the second equation in (3) becomes u u tt u t. CLASSIFICATION BY ORDER The order of a differential equation (either ODE or PDE) is the order of the highest derivative in the equation. For eample, second order d d 5( first order ) 3 4 e d is a second-order ordinar differential equation. First-order ordinar differential equations are occasionall written in differential form M(, ) d N(, ) 0. For eample, if we assume that denotes the dependent variable in ( ) d 4 0, then d, so b dividing b the differential d, we get the alternative form 4. See the Remarks at the end of this section. 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 of n variables:,,,..., (n). For both practical and theoretical reasons we shall also make the assumption hereafter that it is possible to solve an ordinar differential equation in the form (4) uniquel for the * Ecept for this introductor section, onl ordinar differential equations are considered in A First Course in Differential Equations with Modeling Applications, Ninth Edition. In that tet the word equation and the abbreviation DE refer onl to ODEs. Partial differential equations or PDEs are considered in the epanded volume Differential Equations with Boundar-Value Problems, Seventh Edition.

4 CHAPTER INTRODUCTION TO DIFFERENTIAL EQUATIONS highest derivative (n) in terms of the remaining n variables. The differential equation d n, (5) d f (,,,..., n (n ) ) 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 d f (, ) and d f (,, ) d to represent general first- and second-order ordinar differential equations. For eample, the normal form of the first-order equation 4 is ( ) 4; the normal form of the second-order equation 6 0 is 6. See the Remarks. CLASSIFICATION BY LINEARITY An nth-order ordinar differential equation (4) is said to be linear if F is linear in,,..., (n). This means that an nth-order ODE is linear when (4) is a n () (n) a n () (n ) a () a 0 () g() 0 or a n () d n. (6) d a n () d n n d a () n d a 0() g() Two important special cases of (6) are linear first-order (n ) and linear secondorder (n ) DEs: a (). (7) d a 0() g() and a () d d a () d a 0() g() In the additive combination on the left-hand side of equation (6) we see that the characteristic two properties of a linear ODE are as follows: The dependent variable and all its derivatives,,..., (n) are of the first degree, that is, the power of each term involving is. The coefficients a 0, a,..., a n of,,..., (n) depend at most on the independent variable. The equations ( )d 4 0, 0, and d 3 5 e 3 d d are, in turn, linear first-, second-, and third-order ordinar differential equations. We have just demonstrated that the first equation is linear in the variable b writing it in the alternative form 4. A nonlinear ordinar differential equation is simpl one that is not linear. Nonlinear functions of the dependent variable or its derivatives, such as sin or, cannot appear in a linear equation. Therefore e nonlinear term: coefficient depends on nonlinear term: nonlinear function of nonlinear term: power not d ( ) e, d sin 0, and 4 d 0 d 4 are eamples of nonlinear first-, second-, and fourth-order ordinar differential equations, respectivel. SOLUTIONS As was stated before, one of the goals in this course is to solve, or find solutions of, differential equations. In the net definition we consider the concept of a solution of an ordinar differential equation.

. DEFINITIONS AND TERMINOLOGY 5 DEFINITION.. Solution of an ODE An function, defined on an interval I and possessing at least 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 that possesses at least n derivatives and for which F(, (), (),..., (n) ()) 0 for all in I. We sa that satisfies the differential equation on I. For our purposes we shall also assume that a solution is a real-valued function. In our introductor discussion we saw that e 0. is a solution of d 0. on the interval (, ). Occasionall, it will be convenient to denote a solution b the alternative smbol (). INTERVAL OF DEFINITION You cannot think solution of an ordinar differential equation without simultaneousl thinking interval. The interval I in Definition.. is variousl called the interval of definition, the interval of eistence, 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, ), and so on. EXAMPLE Verification of a Solution Verif that the indicated function is a solution of the given differential equation on the interval (, ). (a) >d / ; 6 4 left-hand side: right-hand side: / 6 4 / 4 4 3, we see that each side of the equation is the same for ever real number. Note that / 4 is, b definition, the nonnegative square root of. left-hand side: (e e ) (e e ) e 0, right-hand side: 0. (b) 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. (a) From d 6 (4 3 ) 4 3, 6 4 (b) From the derivatives e e and e e we have, for ever real number, Note, too, that in Eample each differential equation possesses the constant solution 0,. A solution of a differential equation that is identicall zero on an interval I is said to be a trivial solution. SOLUTION CURVE The graph of a solution of an ODE is called a solution curve. Since is a differentiable function, it is continuous on its interval I of definition. Thus there ma be a difference between the graph of the function and the

6 CHAPTER INTRODUCTION TO DIFFERENTIAL EQUATIONS graph of the solution. Put another wa, the domain of the function need not be the same as the interval I of definition (or domain) of the solution. Eample illustrates the difference. EXAMPLE Function versus Solution (a) function /, 0 The domain of, considered simpl as a function, is the set of all real numbers ecept 0. When we graph, we plot points in the -plane corresponding to a judicious 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. Now is also a solution of the linear first-order differential equation 0. (Verif.) But when we sa that is a solution of this DE, we mean that 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 that does not contain 0, such as ( 3, ), (, 0), (, 0), or (0, ). Because the solution curves defined b for 3 and 0 are simpl segments, or pieces, of the solution curves defined b for 0 and 0, respectivel, it makes sense to take the interval I to be as large as possible. Thus we take I to be either (, 0) or (0, ). The solution curve on (0, ) is shown in Figure..(b). (b) solution /, (0, ) FIGURE.. The function is not the same as the solution EXPLICIT AND IMPLICIT SOLUTIONS You should be familiar with the terms eplicit functions and implicit functions from our stu of calculus. A solution in which the dependent variable is epressed solel in terms of the independent variable and constants is said to be an eplicit solution. For our purposes, let us think of an eplicit solution as an eplicit formula () that we can manipulate, evaluate, and differentiate using the standard rules. We have just seen in the last two eamples that, e, and are, in turn, eplicit solutions of d / 6 4, 0, and 0. Moreover, the trivial solution 0 is an eplicit solution of all three equations. When we get down to the business of actuall solving some ordinar differential equations, ou will see that methods of solution do not alwas lead directl to an eplicit solution (). This is particularl true when we attempt to solve nonlinear first-order differential equations. Often we have to be content with a relation or epression G(, ) 0 that defines a solution implicitl. DEFINITION..3 Implicit Solution of an ODE A relation G(, ) 0 is said to be an implicit solution of an ordinar differential equation (4) on an interval I, provided that there eists at least one function that satisfies the relation as well as the differential equation on I. It is beond the scope of this course to investigate the conditions under which a relation G(, ) 0 defines a differentiable function. 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 that satisfies both the relation (that is, G(, ()) 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 of and obtain one or more eplicit solutions. See the Remarks.

. DEFINITIONS AND TERMINOLOGY 7 5 EXAMPLE 3 Verification of an Implicit Solution 5 The relation 5 is an implicit solution of the differential equation d on the open interval ( 5, 5). B implicit differentiation we obtain (8) (a) implicit solution 5 d. d d d d 5 or d d 0 5 5 Solving the last equation for the smbol d gives (8). Moreover, solving 5 for in terms of ields 5. The two functions () 5 and () 5 satisf the relation (that is, 5 and 5) and are eplicit solutions defined on the interval ( 5, 5). The solution curves given in Figures..(b) and..(c) are segments of the graph of the implicit solution in Figure..(a). (b) eplicit solution 5, 5 5 5 5 (c) eplicit solution 5 5, 5 5 FIGURE.. An implicit solution and two eplicit solutions of c>0 c=0 c<0 FIGURE..3 Some solutions of sin An relation of the form 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, if c 5, we cannot sa that 5 0 is an implicit solution of the equation. (Wh not?) Because the distinction between an eplicit solution and an implicit solution should be intuitivel clear, we will not belabor the issue b alwas saing, Here is an eplicit (implicit) solution. FAMILIES OF SOLUTIONS The stu of differential equations is similar to that of integral calculus. In some tets a solution is sometimes referred to as an integral of the equation, and its graph is called an integral curve. When evaluating an antiderivative or indefinite integral in calculus, we use a single constant c of integration. Analogousl, when solving a first-order differential equation F(,, ) 0, we usuall obtain a solution containing a single arbitrar constant or parameter 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,..., c n ) 0. This means that a single differential equation can possess an infinite number of solutions corresponding to the unlimited number of choices for the parameter(s). A solution of a differential equation that is free of arbitrar parameters is called a particular solution. For eample, the one-parameter famil c cos is an eplicit solution of the linear first-order equation sin on the interval (, ). (Verif.) Figure..3, obtained b using graphing software, shows the graphs of some of the solutions in this famil. The solution cos, the blue curve in the figure, is a particular solution corresponding to c 0. Similarl, on the interval (, ), c e c e is a two-parameter famil of solutions of the linear second-order equation 0 in Eample. (Verif.) Some particular solutions of the equation are the trivial solution 0 (c c 0), e (c 0, c ), 5e e (c 5, c ), and so on. Sometimes a differential equation possesses a solution that is not a member of a famil of solutions of the equation that is, a solution that cannot be obtained b specializing an of the parameters in the famil of solutions. Such an etra solution is called a singular solution. For eample, we have seen that and 0 are solutions of the differential equation d / 6 4 on (, ). In Section. we shall demonstrate, b actuall solving it, that the differential equation d / possesses the oneparameter famil of solutions. When c 0, the resulting particular solution is. But notice that the trivial solution 0 is a singular solution, since 6 4 ( 4 c)

8 CHAPTER INTRODUCTION TO DIFFERENTIAL EQUATIONS it is not a member of the famil ( 4 c) ; there is no wa of assigning a value to the constant c to obtain 0. In all the preceding eamples we 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 and the dependent variable b. EXAMPLE 4 Using Different Smbols The functions c cos 4t and c sin 4t, where c and c are arbitrar constants or parameters, are both solutions of the linear differential equation 6 0. For c cos 4t the first two derivatives with respect to t are 4c sin 4t and 6c cos 4t. Substituting and then gives 6 6c cos 4t 6(c cos 4t) 0. In like manner, for c sin 4t we have 6c sin 4t, and so 6 6c sin 4t 6(c sin 4t) 0. Finall, it is straightforward to verif that the linear combination of solutions, or the two-parameter famil c cos 4t c sin 4t, is also a solution of the differential equation. The net eample shows that a solution of a differential equation can be a piecewise-defined function. EXAMPLE 5 A Piecewise-Defined Solution c = c = You should verif that the one-parameter famil c 4 is a one-parameter famil of solutions of the differential equation 4 0 on the inverval (, ). See Figure..4(a). The piecewise-defined differentiable function 4, 0 4, 0 FIGURE..4 4 0 (a) two eplicit solutions c =, < 0 c =, 0 (b) piecewise-defined solution Some solutions of is a particular solution of the equation but cannot be obtained from the famil c 4 b a single choice of c; the solution is constructed from the famil b choosing c for 0 and c for 0. See Figure..4(b). SYSTEMS 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. For eample, if and denote dependent variables and t denotes the independent variable, then a sstem of two first-order differential equations is given b d f(t,, ) dt g(t,, ). dt (9)

. DEFINITIONS AND TERMINOLOGY 9 A solution of a sstem such as (9) is a pair of differentiable functions (t), (t), defined on a common interval I, that satisf each equation of the sstem on this interval. REMARKS (i) A few last words about implicit solutions of differential equations are in order. In Eample 3 we were able to solve the relation 5 for in terms of to get two eplicit solutions, () 5 and () 5, of the differential equation (8). But don t read too much into this one eample. Unless it is eas or important or ou 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 that is a solution of a DE, et we might not be able to solve G(, ) 0 using analtical methods such as algebra. The solution curve of ma be a segment or piece of the graph of G(, ) 0. See Problems 45 and 46 in Eercises.. Also, read the discussion following Eample 4 in Section.. (ii) Although the concept of a solution has been emphasized in this section, ou should also be aware that a DE does not necessaril have to possess a solution. See Problem 39 in Eercises.. The question of whether a solution eists will be touched on in the net section. (iii) It might not be apparent whether a first-order ODE written in differential form M(, )d N(, ) 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.. (iv) It might not seem like a big deal to assume that F(,,,..., (n) ) 0 can be solved for (n), but one should be a little bit careful here. There are eceptions, and there certainl are some problems connected with this assumption. See Problems 5 and 53 in Eercises.. (v) You ma run across the term closed form solutions in DE tets or in lectures in courses in differential equations. Translated, this phrase usuall refers to eplicit solutions that are epressible in terms of elementar (or familiar) functions: finite combinations of integer powers of, roots, eponential and logarithmic functions, and trigonometric and inverse trigonometric functions. (vi) 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,...,c n ) 0 b appropriate choices of the parameters c i, i,,..., 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 one can be assured that not onl does a solution eist on an interval but also that a famil of solutions ields all possible solutions. Nonlinear ODEs, with the eception of some first-order equations, are usuall difficult or impossible to solve in terms of elementar functions. Furthermore, if we happen to obtain a famil of solutions for a nonlinear equation, it is not obvious whether this famil contains all solutions. On a practical level, then, the designation general solution is applied onl to linear ODEs. Don t be concerned about this concept at this point, but store the words general solution in the back of our mind we will come back to this notion in Section.3 and again in Chapter 4.

0 CHAPTER INTRODUCTION TO DIFFERENTIAL EQUATIONS EXERCISES. In Problems 8 state the order of the given ordinar differential equation. Determine whether the equation is linear or nonlinear b matching it with (6).. ( ) 4 5 cos. 3. t 5 (4) t 3 6 0 4. 5. 6. 7. (sin ) (cos ) 8. d3 d 3 d 4 0 d u du u cos(r u) dr dr d d B d d R dt k R ẍ ẋ 3 ẋ 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. ( ) 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 the given differential equation. Assume an appropriate interval I of definition for each solution.. 0; e /. 0 4; dt 6 5 6 5 e 0t 3. 6 3 0; e 3 cos 4. tan ; (cos )ln(sec tan ) In Problems 5 8 verif that the indicated function () is an eplicit solution of the given first-order differential equation. Proceed as in Eample, b considering simpl as a function, give its domain. Then b considering as a solution of the differential equation, give at least one interval I of definition. 5. ( ) 8; 4 Answers to selected odd-numbered problems begin on page ANS-. 6. 5 ; 5 tan 5 7. ; (4 ) 8. 3 cos ; ( sin ) / In Problems 9 and 0 verif that the indicated epression is an implicit solution of the given first-order differential equation. Find at least one eplicit solution () in each case. Use a graphing utilit to obtain the graph of an eplicit solution. Give an interval I of definition of each solution. 9. 0. d ( ) 0; In Problems 4 verif that the indicated famil of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution... 3. 4. dx dt (X )( X); ln X X t dp P( P); P c e t dt c e t d ; e e t dt c e 0 d 4 d d 4 0; c e c e d 3 3 d d 3 d d ; c c c 3 ln 4 5. Verif that the piecewise-defined function, 0, 0 is a solution of the differential equation 0 on (, ). 6. In Eample 3 we saw that () 5 and () 5 are solutions of d on the interval ( 5, 5). Eplain wh the piecewisedefined function 5, 5, 5 0 0 5 is not a solution of the differential equation on the interval ( 5, 5).

. DEFINITIONS AND TERMINOLOGY In Problems 7 30 find values of m so that the function e m is a solution of the given differential equation. 7. 0 8. 5 9. 5 6 0 30. 7 4 0 In Problems 3 and 3 find values of m so that the function m is a solution of the given differential equation. 3. 0 3. 7 5 0 In Problems 33 36 use the concept that c,, is a constant function if and onl if 0 to determine whether the given differential equation possesses constant solutions. 33. 3 5 0 34. 3 35. ( ) 36. 4 6 0 In Problems 37 and 38 verif that the indicated pair of functions is a solution of the given sstem of differential equations on the interval (, ). d d 37. 3 38. 4 et dt dt d 5 3; dt 4 et ; dt e t 3e 6t, cos t sin t e t 5e 6t 5 et cos t sin t 5 et Discussion Problems 39. Make up a differential equation that does not possess an real solutions. 40. Make up a differential equation that ou feel confident possesses onl the trivial solution 0. Eplain our reasoning. 4. What function do ou know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple k of itself? Write each answer in the form of a first-order differential equation with a solution. 4. What function (or functions) do ou know from calculus is such that its second derivative is itself? Its second derivative is the negative of itself? Write each answer in the form of a second-order differential equation with a solution., 43. Given that sin is an eplicit solution of the first-order differential equation. Find d an interval I of definition. [Hint: I is not the interval (, ).] 44. Discuss wh it makes intuitive sense to presume that the linear differential equation 4 5 sin t has a solution of the form A sin t B cos t, where A and B are constants. Then find specific constants A and B so that A sin t B cos t is a particular solution of the DE. In Problems 45 and 46 the given figure represents the graph of an implicit solution G(, ) 0 of a differential equation d f (, ). In each case the relation G(, ) 0 implicitl defines several solutions of the DE. Carefull reproduce each figure on a piece of paper. Use different colored pencils to mark off segments, or pieces, on each graph that correspond to graphs of solutions. Keep in mind that a solution must be a function and differentiable. Use the solution curve to estimate an interval I of definition of each solution. 45. 46. FIGURE..5 Graph for Problem 45 FIGURE..6 Graph for Problem 46 47. The graphs of 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 d (3 3 ) ( 3 3 ).

CHAPTER INTRODUCTION TO DIFFERENTIAL EQUATIONS 48. The graph in Figure..6 is the member of the famil of folia in Problem 47 corresponding to c. Discuss: How can the DE in Problem 47 help in finding points on the graph of 3 3 3 where the tangent line is vertical? How does knowing where a tangent line is vertical help in determining an interval I of definition of a solution of the DE? Carr out our ideas, and compare with our estimates of the intervals in Problem 46. 49. In Eample 3 the largest interval I over which the eplicit solutions () and () are defined is the open interval ( 5, 5). Wh can t the interval I of definition be the closed interval [ 5, 5]? 50. In Problem a one-parameter famil of solutions of the DE P P( P) is given. Does an solution curve pass through the point (0, 3)? Through the point (0, )? 5. Discuss, and illustrate with eamples, how to solve differential equations of the forms d f () and d d f (). 5. The differential equation ( ) 4 3 0 has the form given in (4). Determine whether the equation can be put into the normal form d f (, ). 53. The normal form (5) of an nth-order differential equation is equivalent to (4) whenever both forms have eactl the same solutions. Make up a first-order differential equation for which F(,, ) 0 is not equivalent to the normal form d f (, ). 54. Find a linear second-order differential equation F(,,, ) 0 for which c c is a twoparameter famil of solutions. Make sure that our equation is free of the arbitrar parameters c and c. Qualitative information about a solution () of a differential equation can often be obtained from the equation itself. Before working Problems 55 58, recall the geometric significance of the derivatives d and d d. 55. Consider the differential equation >d e. (a) Eplain wh a solution of the DE must be an increasing function on an interval of the -ais. (b) What are lim >d and lim >d? What does : : this suggest about a solution curve as :? (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution () of the differential equation whose shape is suggested b parts (a) (c). 56. Consider the differential equation d 5. (a) Either b inspection or b the method suggested in Problems 33 36, find a constant solution of the DE. (b) Using onl the differential equation, find intervals on the -ais on which a nonconstant solution () is increasing. Find intervals on the -ais on which () is decreasing. 57. Consider the differential equation d (a b), where a and b are positive constants. (a) Either b inspection or b the method suggested in Problems 33 36, find two constant solutions of the DE. (b) Using onl the differential equation, find intervals on the -ais on which a nonconstant solution () is increasing. Find intervals on which () is decreasing. (c) Using onl the differential equation, eplain wh a b is the -coordinate of a point of inflection of the graph of a nonconstant solution (). (d) On the same coordinate aes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the -plane into three regions. In each region, sketch the graph of a nonconstant solution () whose shape is suggested b the results in parts (b) and (c). 58. Consider the differential equation 4. (a) Eplain wh there eist no constant solutions of the DE. (b) Describe the graph of a solution (). 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. (d) Sketch the graph of a solution () of the differential equation whose shape is suggested b parts (a) (c). Computer Lab Assignments In Problems 59 and 60 use a CAS to compute all derivatives and to carr out the simplifications needed to verif that the indicated function is a particular solution of the given differential equation. 59. (4) 0 58 580 84 0; e 5 cos 60. 3 0 78 0; cos(5 ln ) sin(5 ln ) 0 3