4.317 d 4 y. 4 dx d 2 y dy. 20. dt d 2 x. 21. y 3y 3y y y 6y 12y 8y y (4) y y y (4) 2y y 0. d 4 y 26.

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1 38 CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS sstems are also able, b means of their dsolve commands, to provide eplicit solutions of homogeneous linear constant-coefficient differential equations. In the classic tet Differential Equations b Ralph Palmer Agnew * (used b the author as a student) the following statement is made: It is not reasonable to epect students in this course to have computing skill and equipment necessar for efficient solving of equations such as 4.37 d 4 (3) d.79 d 3 4 d.46 d d d d Although it is debatable whether computing skills have improved in the intervening ears, it is a certaint that technolog has. If one has access to a computer algebra sstem, equation (3) could now be considered reasonable. After simplification and some relabeling of output, Mathematica ields the (approimate) general solution c e.7885 cos(.6865) c e.7885 sin(.6865) c 3 e cos(.7598) c 4 e sin(.7598). Finall, if we are faced with an initial-value problem consisting of, sa, a fourth-order equation, then to fit the general solution of the DE to the four initial conditions, we must solve four linear equations in four unknowns (the c, c, c 3, c 4 in the general solution). Using a CAS to solve the sstem can save lots of time. See Problems 59 and 6 in Eercises 4.3 and Problem 35 in Chapter 4 in Review. * McGraw-Hill, New York, 96. EXERCISES 4.3 In Problems 4 find the general solution of the given second-order differential equation Answers to selected odd-numbered problems begin on page ANS-4. d 3. d (4) 4. (4) 5. 6 d 4 d 4 d 9 4 d d 4 6. d 7 d 8 4 d In Problems 5 8 find the general solution of the given higher-order differential equation. 7. d 5 u dr 5 d 4 u 5 dr d 3 u 4 dr d u du 3 dr dr 5u d 3 u d u u 3 8. In Problems 9 36 solve the given initial-value problem. 9. 6, (), () 3. d 5 ds 5 7 d 4 ds 4 d 3 ds 3 8 d ds d, d 3, 3

2 4.3 HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS d d 4 5, (), () , (), () 5 33., () () 34., () 5, () , (), (), () , () (), () In Problems 37 4 solve the given boundar-value problem , (), () 38. 4, (), (p) 39., (), 4., (), (p) In Problems 4 and 4 solve the given problem first using the form of the general solution given in (). Solve again, this time using the form given in (). FIGURE Graph for Problem FIGURE Graph for Problem , (), () 5 4., (), () In Problems each figure represents the graph of a particular solution of one of the following differential equations: (a) 34 (b) 4 (c) (d) (e) (f) 3 Match a solution curve with one of the differential equations. Eplain our reasoning. 43. π FIGURE Graph for Problem π FIGURE Graph for Problem 48 FIGURE 4.3. Graph for Problem FIGURE Graph for Problem 44 Discussion Problems 49. The roots of a cubic auiliar equation are m 4 and m m 3 5. What is the corresponding homogeneous linear differential equation? Discuss: Is our answer unique? 5. Two roots of a cubic auiliar equation with real coefficients are m and m 3 i. What is the corresponding homogeneous linear differential equation?

3 4.6 VARIATION OF PARAMETERS 6 The first n equations in this sstem, like u u in (4), are assumptions that are made to simplif the resulting equation after p u () () u n () n () is substituted in (9). In this case Cramer s rule gives u k W k, k,,..., n, W where W is the Wronskian of,,..., n and W k is the determinant obtained b replacing the kth column of the Wronskian b the column consisting of the righthand side of () that is, the column consisting of (,,..., f ()). When n, we get (5). When n 3, the particular solution is p u u u 3 3, where,, and 3 constitute a linearl independent set of solutions of the associated homogeneous DE and u, u, u 3 are determined from u W W, u W W, u 3 W 3 W, () W p f () 3 3 p, 3 W p f () 3 3 p, W 3 p 3 f () p, and W p 3 3 p. 3 See Problems 5 and 6 in Eercises 4.6. REMARKS (i) Variation of parameters has a distinct advantage over the method of undetermined coefficients in that it will alwas ield a particular solution p provided that the associated homogeneous equation can be solved. The present method is not limited to a function f () that is a combination of the four tpes listed on page 4. As we shall see in the net section, variation of parameters, unlike undetermined coefficients, is applicable to linear DEs with variable coefficients. (ii) In the problems that follow, do not hesitate to simplif the form of p. Depending on how the antiderivatives of u and u are found, ou might not obtain the same p as given in the answer section. For eample, in Problem 3 in Eercises 4.6 both p sin cos and p 4 sin cos are valid answers. In either case the general solution c p simplifies to c cos c sin cos. Wh? EXERCISES 4.6 In Problems 8 solve each differential equation b variation of parameters.. sec. tan 3. sin 4. sec u tan u 5. cos 6. sec 7. cosh 8. sinh 9. 4 e. 9 9 e 3 Answers to selected odd-numbered problems begin on page ANS e e sin e 4. e t arctan t 5. e t ln t e sec e /

4 6 CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS In Problems 9 solve each differential equation b variation of parameters, subject to the initial conditions (), (). 9. 4e /.. 8 e e. 44 ( 6)e In Problems 3 and 4 the indicated functions are known linearl independent solutions of the associated homogeneous differential equation on (, ). Find the general solution of the given nonhomogeneous equation. 3. ( 4) 3/ ; / cos, / sin 4. sec(ln ); cos(ln ), sin(ln ) In Problems 5 and 6 solve the given third-order differential equation b variation of parameters. 5. tan 6. 4sec Discussion Problems In Problems 7 and 8 discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve the given differential equation. Carr out our ideas sin e tan e 9. What are the intervals of definition of the general solutions in Problems, 7, 9, and 8? Discuss wh the interval of definition of the general solution in Problem 4 is not (, ). 3. Find the general solution of given that is a solution of the associated homogeneous equation. 3. Suppose p () u () () u () (), where u and u are defined b (5) is a particular solution of () on an interval I for which P, Q, and f are continuous. Show that p can be written as p () where and are in I, G(, t)f(t), G(, t) (t) () () (t), W(t) () (3) and W(t) W( (t), (t)) is the Wronskian. The function G(, t) in (3) is called the Green s function for the differential equation (). 3. Use (3) to construct the Green s function for the differential equation in Eample 3. Epress the general solution given in (8) in terms of the particular solution (). 33. Verif that () is a solution of the initial-value problem d Pd d d Q f(), ( ), ( ). on the interval I. [Hint: Look up Leibniz s Rule for differentiation under an integral sign.] 34. Use the results of Problems 3 and 33 and the Green s function found in Problem 3 to find a solution of the initial-value problem e, (), () using (). Evaluate the integral. 4.7 CAUCHY-EULER EQUATION REVIEW MATERIAL Review the concept of the auiliar equation in Section 4.3. INTRODUCTION The same relative ease with which we were able to find eplicit solutions of higher-order linear differential equations with constant coefficients in the preceding sections does not, in general, carr over to linear equations with variable coefficients. We shall see in Chapter 6 that when a linear DE has variable coefficients, the best that we can usuall epect is to find a solution in the form of an infinite series. However, the tpe of differential equation that we consider in this section is an eception to this rule; it is a linear equation with variable coefficients whose general solution can alwas be epressed in terms of powers of, sines, cosines, and logarithmic functions. Moreover, its method of solution is quite similar to that for constant-coefficient equations in that an auiliar equation must be solved.

5 ANS-4 ANSWERS FOR SELECTED ODD-NUMBERED PROBLEMS ANSWERS FOR SELECTED ODD-NUMBERED PROBLEMS CHAPTER 4 EXERCISES 3.3 (PAGE ). (t) e t 3. 5,, 47 das. The time when (t) and z(t) are the same makes sense because most of A and half of B are gone, so half of C should have been formed (a) 3. d t 3 t (b) (t) (t) 5; (3) 47.4 lb 5. i() i, s() n i, r() CHAPTER 3 IN REVIEW (PAGE 3). dp.5p 3. P(45) 8.99 billion (a) 9. d d (b). (t) ac e ak t c e, (t) c ak t ( c e akt ) k /k 3. c e 5. (a) (t) (e t e t ) z(t) d 3 t t L di (R R )i R i 3 e t e t L di 3 R i (R R 3 ) i 3 E(t) ln BT T B, BT T B T(t) BT T B T T B ek(b)t i(t) 4t 5 t, t, t E(t) p() r()g Kq() d (b) The ratio is increasing; the ratio is constant. Kp (d) r() gk q() d ; r() Kp B(CKp bg) EXERCISES 4. (PAGE 8). e e ln 9. (, ). (a) e (b) sinh e (e e ) sinh 3. (a) e cos e sin (b) no solution (c) e cos e p/ e sin (d) c e sin, where c is arbitrar 5. dependent 7. dependent 9. dependent. independent 3. The functions satisf the DE and are linearl independent on the interval since W(e 3, e 4 ) 7e ; c e 3 c e The functions satisf the DE and are linearl independent on the interval since W(e cos, e sin ) e ; c e cos c e sin. 7. The functions satisf the DE and are linearl independent on the interval since W( 3, 4 ) 6 ; c 3 c The functions satisf the DE and are linearl independent on the interval since W(,, ln ) 9 6 ; c c c 3 ln. 35. (b) p 3 3e ; p 6 3 e EXERCISES 4. (PAGE 3). e 3. sin 4 5. sinh 7. e / ln. 3. cos (ln ) e, p 9. e, p 5 e3 EXERCISES 4.3 (PAGE 38). c c e /4 3. c e 3 c e 5. c e 4 c e 4 7. c e /3 c e /4 9. c cos 3 c sin 3. e (c cos c sin ) 3. e /3 (c cos 5. c c e 3 c sin c 3 e 5 3 ) 7. c e c e 3 c 3 e 3 9. u c e t e t (c cos t c 3 sin t). c e c e c 3 e 3. c c e / (c 3 cos 3 c 5. c cos 3 c sin 3 4 sin 3 ) c 3 cos 3 c 4 sin 3 7. u c e r c re r c 3 e r c 4 re r c 5 e 5r 9. cos 4 3. sin 4 3 e(t) 3 e5(t) 33.

6 ANSWERS FOR SELECTED ODD-NUMBERED PROBLEMS ANS e6 6 e6 35. c e 3 c e e 5 e c c e c e c e e3 3 5 e3 ; 4. c c c 3 e c e 3 c e 4 7 e4 cosh sinh3 45. c e c e 3 e c cos 5 c sin 5 sin c e 3 c e 3 49 e4 343 e4 EXERCISES 4.4 (PAGE 48) 5. c e c e 6 3 e 4 e 4 e 5. c e c e 53. e 3 (c cos c sin ) 3 e sin 3. c e 5 c e c cos 5 c sin 5 cos c e c e e 7. c cos 3 c / c sin 3 ( )e 3 cos 3 c sin 3 9. c c e sin cos cos 3. c e / c e / e / 59. c c c 3 e c cos c sin 3 4 cos 6. c e c e c 3 e 6 3 e c cos c sin cos sin c e cos c e sin 4 e sin c e c e cos 63. c c c 3 e c 4 e e e8 5 8 e e sin 5 cos 69. cos 3 sin 8 3 cos cos. c c c 3 e cos 37 sin 7. e cos 3 64 e sin c e c e c 3 e e 5. c cos c sin c 3 cos c 4 sin 3 EXERCISES 4.6 (PAGE 6). c cos c sin sin cos ln cos 7. sin 9. e / c 3 cos c sin cos 3. e cos 9e sin 7e 4 5. c cos c sin 6 cos 33. F t F 7. c e c e sinh t cos t 9. c 35. e 9e e e c e 4 e 4t e ln e e5 t, cos 6(cot ) sin sin 3. c e c e (e e ) ln( e ) sin 3 3 cos 3 3. c e c e e sin e 4. cos 5 sin 6 3 sin, > 3 cos 5 6 sin, > EXERCISES 4.5 (PAGE 56). (3D )(3D ) sin 3. (D 6)(D ) 6 5. D(D 5) e 7. (D )(D )(D 5) e 9. D(D )(D D 4) 4 5. D 4 7. D(D ) 9. D 4. D 3 (D 6) 3. (D )(D ) 3 5. D(D D 5) 7.,,, 3, 4 9. e 6, e 3/ 3. cos 5, sin 5 33., e 5, e 5 5. c e t c te t t e t ln t 3 4 t e t 7. c e sin c e cos 3 e sin 9. 4 e/ 3 4 e/ 8 e / 4 e/. 4 9 e e 4 e 9 e 3. c / cos c / sin / 5. 3 e cos ln cos c c cos c 3 sin ln cos sin ln sec tan EXERCISES 4.7 (PAGE 68). c c 3. c c ln 5. c cos( ln ) c sin( ln ) ANSWERS FOR SELECTED ODD-NUMBERED PROBLEMS CHAPTER 4