FIRST- AND SECOND-ORDER IVPS The problem given in (1) is also called an nth-order initial-value problem. For example, Solve: Solve:

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1 .2 INITIAL-VALUE PROBLEMS 3.2 INITIAL-VALUE PROBLEMS REVIEW MATERIAL Normal form of a DE Solution of a DE Famil of solutions INTRODUCTION We are often interested in problems in which we seek a solution () of a differential equation so that () satisfies prescribed side conditions that is, conditions imposed on the unknown () or its derivatives. On some interval I containing 0 the problem Solve: d n d n f,,,..., (n ) Subject to: ( 0 ) 0, ( 0 ),..., (n ) ( 0 ) n, () where 0,,..., n are arbitraril specified real constants, is called an initial-value problem (IVP). The values of () and its first n derivatives at a single point 0, ( 0 ) 0, ( 0 ),..., (n ) ( 0 ) n, are called initial conditions. solutions of the DE ( 0, 0 ) FIRST- AND SECOND-ORDER IVPS The problem given in () is also called an nth-order initial-value problem. For eample, Solve: d f (, ) d Subject to: ( 0 ) 0 (2) I FIGURE.2. Solution of first-order IVP solutions of the DE m = ( 0, 0 ) I FIGURE.2.2 Solution of second-order IVP and Solve: d 2 d 2 f (,, ) Subject to: ( 0 ) 0, ( 0 ) 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 f (, ) on an interval I containing 0 so that its graph passes through the specified point ( 0, 0 ). A solution curve is shown in blue in Figure.2.. For (3) we want to find a solution () of the differential equation f (,, ) on an interval I containing 0 so that its graph not onl passes through ( 0, 0 ) but the slope of the curve at this point is the number. A solution curve is shown in blue in Figure.2.2. The words initial conditions derive from phsical sstems where the independent variable is time t and where (t 0 ) 0 and (t 0 ) represent the position and velocit, respectivel, of an object at some beginning, or initial, time t 0. Solving an nth-order initial-value problem such as () frequentl entails first finding an n-parameter famil of solutions of the given differential equation and then using the n initial conditions at 0 to determine numerical values of the n constants in the famil. The resulting particular solution is defined on some interval I containing the initial point 0. (3) EXAMPLE Two First-Order IVPs In Problem 4 in Eercises. ou were asked to deduce that ce is a oneparameter famil of solutions of the simple first-order equation. All the solutions in this famil are defined on the interval (, ). If we impose an initial condition, sa, (0) 3, then substituting 0, 3 in the famil determines the

2 4 CHAPTER INTRODUCTION TO DIFFERENTIAL EQUATIONS (0, 3) (, 2) constant 3 ce 0 c. Thus 3e is a solution of the IVP, (0) 3. Now if we demand that a solution curve pass through the point (, 2) rather than (0, 3), then () 2 will ield 2 ce or c 2e. In this case 2e is a solution of the IVP, () 2. The two solution curves are shown in dark blue and dark red in Figure.2.3. FIGURE.2.3 Solutions of two IVPs The net eample illustrates another first-order initial-value problem. In this eample notice how the interval I of definition of the solution () depends on the initial condition ( 0 ) 0. EXAMPLE 2 Interval I of Definition of a Solution (a) function defined for all ecept = ± (0, ) 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. Thus ( 2 ). We 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). Considered as a solution of the differential equation 2 2 0, the interval I of definition of ( 2 ) could be taken to be an interval over which () is defined and differentiable. As can be seen in Figure.2.4(a), the largest intervals on which ( 2 ) is a solution are (, ), (, ), and (, ). 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, differentiable, and contains the initial point 0; the largest interval for which this is true is (, ). See the red curve in Figure.2.4(b). See Problems 3 6 in Eercises.2 for a continuation of Eample 2. EXAMPLE 3 Second-Order IVP (b) solution defined on interval containing = 0 FIGURE.2.4 Graphs of function In Eample 4 of Section. we saw that c cos 4t c 2 sin 4t is a two-parameter and solution of IVP in Eample 2 famil of solutions of 6 0. Find a solution of the initial-value problem 6 0,. (4) 2 2, 2 SOLUTION We first appl ( 2) 2 to the given famil of solutions: c cos 2 c 2 sin 2 2. Since cos 2 and sin 2 0, we find that c 2. We net appl ( 2) to the one-parameter famil (t) 2 cos 4t c 2 sin 4t. Differentiating and then setting t 2 and gives 8 sin 2 4c 2 cos 2, from which we see that c 2. Hence 2 cos 4t 4 4 sin 4t is a solution of (4). EXISTENCE AND UNIQUENESS Two fundamental questions arise in considering an initial-value problem: Does a solution of the problem eist? If a solution eists, is it unique?

3 .2 INITIAL-VALUE PROBLEMS For the first-order initial-value problem (2) we ask: Eistence Does the differential equation d d f (, ) possess solutions? { Do an of the solution curves pass through the point ( 0, 0 )? Uniqueness When can we be certain that there is precisel one solution curve { passing through the point ( 0, 0 )? 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 problem. The net eample illustrates an initialvalue problem with two solutions. = 0 FIGURE.2. of the same IVP = 4 /6 (0, 0) Two solutions EXAMPLE 4 An IVP Can Have Several Solutions Each of the functions 0 and satisfies the differential equation d d /2 6 4 and the initial condition (0) 0, so the initial-value problem d d /2, (0) 0 has at least two solutions. As illustrated in Figure.2., 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 problems will probabl be unique. Real life, however, is not so idllic. Therefore it is desirable to 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 first-order 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 4 to address the question of eistence and uniqueness of a second-order initial-value problem. THEOREM.2. Eistence of a Unique Solution d R c a FIGURE.2.6 ( 0, 0 ) I 0 b Rectangular region 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 interval I 0 : ( 0 h, 0 h), h 0, contained in [a, b], and a unique function (), defined on I 0, that is a solution of the initialvalue problem (2). The foregoing result is one of the most popular eistence and uniqueness theorems for first-order differential equations because the criteria of continuit of f (, ) and f are relativel eas to check. The geometr of Theorem.2. is illustrated in Figure.2.6. EXAMPLE 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 f (, ) /2 and f 2 /2

4 6 CHAPTER INTRODUCTION TO DIFFERENTIAL EQUATIONS 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 centered at 2 on which the initial-value problem d d /2, (2) has a unique solution. 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 entire -plane. It can be further shown that the interval I on which each solution is defined is (, ). INTERVAL OF EXISTENCE/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 (), the interval I over which the solution () is defined or eists, and the interval I 0 of eistence and uniqueness. Eample 2 of Section. illustrated 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 R b itself is sufficient to guarantee the eistence of at least one solution of d d f (, ), ( 0 ) 0, defined 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 on some interval I 0 containing 0, but it is the onl solution satisfing ( 0 ) 0. However, Theorem.2. does not give an indication of the sizes of intervals I and I 0 ; the interval 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, 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 44 in Eercises.2. REMARKS (i) The conditions in Theorem.2. are sufficient but not necessar. This means that 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 ) is a point interior to R. However, if the conditions stated in the hpothesis of Theorem.2. do not hold, 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 reveals that the hpotheses of Theorem.2. do not hold on the line 0 for the differential equation d d /2, 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. On the other hand, the hpotheses of Theorem.2. do not hold on the line for the differential equation d d. 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 43 in Eercises.2.

5 .2 INITIAL-VALUE PROBLEMS 7 EXERCISES.2 In Problems and 2, ( c e ) is a one-parameter famil of solutions of the first-order DE 2. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition.. (0) 3 2. ( ) 2 In Problems 3 6, ( 2 c) is a one-parameter famil of solutions of the first-order DE Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined. 3. (2) (0) 6. In Problems 7 0, c cos t 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. ( 2) 0, ( 2) In Problems 4, c e c 2 e 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.. ( > 6) 2, ( > 6) 0 ( > 4) 2, ( > 4) 22 (0), (0) 2 2. () 0, () e 3. ( ), ( ) 4. (0) 0, (0) 0 In Problems and 6 determine b inspection at least two solutions of the given first-order IVP.. 3 2/3, (0) , (0) 0 In Problems 7 24 determine a region of the -plane for which the given differential equation would have a unique solution whose graph passes through a point ( 0, 0 ) in the region. d 7. 2/3 8. d ( 2) 2 ( 2) 4 d d Answers to selected odd-numbered problems begin on page ANS-. 9. d d 20. d d 2. (4 2 ) ( 3 ) ( 2 2 ) ( ) In Problems 2 28 determine whether Theorem.2. guarantees that the differential equation 2 9 possesses a unique solution through the given point. 2. (, 4) 26. (, 3) 27. (2, 3) 28. (, ) 29. (a) B inspection find a one-parameter famil of solutions of the differential equation. Verif that each member of the famil is a solution of the initial-value problem, (0) 0. (b) Eplain part (a) b determining a region R in the -plane for which the differential equation would have a unique solution through a point ( 0, 0 ) in R. (c) Verif that the piecewise-defined function 0, 0, 0 satisfies the condition (0) 0. Determine whether this function is also a 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 everwhere, the region R in Theorem.2. can be taken to be the entire -plane. Use the famil of solutions in part (a) to find an eplicit solution of the first-order initial-value problem 2, (0) 0. Even though 0 0 is in the interval ( 2, 2), eplain wh the solution is not defined on this interval. (c) Determine the largest interval I of definition 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, the region R in Theorem.2. can be taken to be the entire -plane. Find a solution from the famil in part (a) that satisfies (0). Then find a solution from the famil in part (a) that satisfies (0). Determine the largest interval I of definition for the solution of each initial-value problem.

6 8 CHAPTER INTRODUCTION TO DIFFERENTIAL EQUATIONS (c) Determine the largest interval I of definition for the solution of the first-order initial-value problem 2, (0) 0. [Hint: The solution is not a member of the famil of solutions in part (a).] (a) Show that a solution from the famil in part (a) of Problem 3 that satisfies 2, (), is (2 ). (b) Then show that a solution from the famil in part (a) of Problem 3 that satisfies 2, (3), is (2 ). (c) Are the solutions in parts (a) and (b) the same? 33. (a) Verif that c is a one-parameter famil of solutions of the differential equation d d 3. (b) B hand, sketch the graph of the implicit solution Find all eplicit solutions () of the DE in part (a) defined b this relation. Give the interval I of definition of each eplicit solution. (c) The point ( 2, 3) is on the graph of , but which of the eplicit solutions in part (b) satisfies ( 2) 3? 34. (a) Use the famil of solutions in part (a) of Problem 33 to find an implicit solution of the initial-value problem d d 3, (2) 4. Then, b hand, sketch the graph of the eplicit solution of this problem and give its interval I of definition. (b) Are there an eplicit solutions of d d 3 that pass through the origin? In Problems 3 38 the graph of a member of a famil of solutions of a second-order differential equation d 2 d 2 f (,, ) is given. Match the solution curve with at least one pair of the following initial conditions. (a) (), () 2 (b) ( ) 0, ( ) 4 (c) (), () 2 (d) (0), (0) 2 (e) (0), (0) 0 (f) (0) 4, (0) 2 3. FIGURE.2.7 Graph for Problem FIGURE.2.8 Graph for Problem 36 FIGURE.2.9 Graph for Problem 37 FIGURE.2.0 Graph for Problem 38 Discussion Problems In Problems 39 and 40 use Problem in Eercises. and (2) and (3) of this section. 39. Find a function f () whose graph at each point (, ) has the slope given b 8e 2 6 and has the -intercept (0, 9). 40. Find a function f () whose second derivative is 2 2 at each point (, ) on its graph and is tangent to the graph at the point corresponding to. 4. Consider the initial-value problem 2, (0) 2. Determine which of the two curves shown in Figure.2. is the onl plausible solution curve. Eplain our reasoning.

7 .3 DIFFERENTIAL EQUATIONS AS MATHEMATICAL MODELS 9 (0, 2) (2, ) (2, ) (a) (b) FIGURE.2.2 Two solutions of the IVP in Problem 44 FIGURE.2. Graphs for Problem Determine a plausible value of 0 for which the graph of the solution of the initial-value problem 2 3 6, ( 0 ) 0 is tangent to the -ais at ( 0, 0). Eplain our reasoning. 43. Suppose that the first-order differential equation d d f (, ) possesses a one-parameter famil of solutions and that f (, ) satisfies the hpotheses of Theorem.2. in some rectangular region R of the -plane. Eplain wh two different solution curves cannot intersect or be tangent to each other at a point ( 0, 0 ) in R. 44. The functions () 6 4, and () 0, 6 4, 0 0 have the same domain but are clearl different. See Figures.2.2(a) and.2.2(b), respectivel. Show that both functions are solutions of the initial-value problem d d /2, (2) on the interval (, ). Resolve the apparent contradiction between this fact and the last sentence in Eample. Mathematical Model 4. Population Growth Beginning in the net section we will see that differential equations can be used to describe or model man different phsical sstems. In this problem suppose that a model of the growing population of a small communit is given b the initial-value problem dp 0.P(t) 20, P(0) 00, dt where P is the number of individuals in the communit and time t is measured in ears. How fast that is, at what rate is the population increasing at t 0? How fast is the population increasing when the population is 00?.3 DIFFERENTIAL EQUATIONS AS MATHEMATICAL MODELS REVIEW MATERIAL Units of measurement for weight, mass, and densit Newton s second law of motion Hooke s law Kirchhoff s laws Archimedes principle INTRODUCTION In this section we introduce the notion of a differential equation as a mathematical model and discuss some specific models in biolog, chemistr, and phsics. Once we have studied some methods for solving DEs in Chapters 2 and 4, we return to, and solve, some of these models in Chapters 3 and. 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. The mathematical description of a sstem of phenomenon is called a mathematical model and is constructed with certain goals in mind. For eample, we ma wish 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 analzing the deca of a radioactive substance either in the fossil or in the stratum in which it was discovered.