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Variation of Parameters 156 ● CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS In Problems 27–34 find linearly independent functions that 55. yЉϩ25y ϭ 20 sin 5x 56. yЉϩy ϭ 4 cos x Ϫ sin x are annihilated by the given differential operator. 57. yЉϩyЈϩy ϭ x sin x 58. yЉϩ4y ϭ cos2x 27. D5 28. D2 ϩ 4D yٞϩ8yЉϭϪ6x2 ϩ 9x ϩ 2 .59 29. D Ϫ D ϩ 30. D2 Ϫ D Ϫ ( 6)(2 3) 9 36 Ϫ yٞϪyЉϩyЈϪy ϭ xex Ϫ e x ϩ 7 .60 31. D2 ϩ 5 32. D2 Ϫ 6D ϩ 10 yٞϪ3yЉϩ3yЈϪy ϭ ex Ϫ x ϩ 16 .61 33. D3 Ϫ 10D2 ϩ 25D 34. D2(D Ϫ 5)(D Ϫ 7) Ϫ 2yٞϪ3yЉϪ3yЈϩ2y ϭ (ex ϩ e x)2 .62 In Problems 35–64 solve the given differential equation by Ϫ ٞϩ Љϭ x ϩ (4) undetermined coefficients 63. y 2y y e 1 64. y(4) Ϫ 4yЉϭ5x2 Ϫ e2x 35. yЉϪ9y ϭ 54 36. 2yЉϪ7yЈϩ5y ϭϪ29 .yЉϩyЈϭ3 38. yٞϩ2yЉϩyЈϭ10 In Problems 65–72 solve the given initial-value problem .37 Љϩ Јϩ ϭ ϩ 39. y 4y 4y 2x 6 65. yЉϪ64y ϭ 16, y(0) ϭ 1, yЈ(0) ϭ 0 40. yЉϩ3yЈϭ4x Ϫ 5 66. yЉϩyЈϭx, y(0) ϭ 1, yЈ(0) ϭ 0 yٞϩyЉϭ8x2 42. yЉϪ2yЈϩy ϭ x3 ϩ 4x .41 67. yЉϪ5yЈϭx Ϫ 2, y(0) ϭ 0, yЈ(0) ϭ 2 43. yЉϪyЈϪ12y ϭ e4x 44. yЉϩ2yЈϩ2y ϭ 5e6x 68. yЉϩ5yЈϪ6y ϭ 10e2x, y(0) ϭ 1, yЈ(0) ϭ 1 45. yЉϪ2yЈϪ3y ϭ 4ex Ϫ 9 Ϫ 69. yЉϩy ϭ 8 cos 2x Ϫ 4 sin x, y(p 2) ϭϪ1, yЈ(p 2) ϭ 0 46. yЉϩ6yЈϩ8y ϭ 3e 2x ϩ 2x > > yٞϪ yЉϩyЈϭxex ϩ y ϭ yЈ ϭ .70 47. yЉϩ25y ϭ 6 sin x 2 5, (0) 2, (0) 2, yЉ(0) ϭϪ1 48. yЉϩ4y ϭ 4 cos x ϩ 3 sin x Ϫ 8 71. yЉϪ4yЈϩ8y ϭ x3, y(0) ϭ 2, yЈ(0) ϭ 4 49. yЉϩ6yЈϩ9y ϭϪxe4x ,y(4) Ϫ yٞϭx ϩ ex, y(0) ϭ 0, yЈ(0) ϭ 0, yЉ(0) ϭ 0 .72 50. yЉϩ3yЈϪ10y ϭ x(ex ϩ 1) yٞ(0) ϭ 0 51. yЉϪy ϭ x2ex ϩ 5 Ϫ 52. yЉϩ2yЈϩy ϭ x2e x Discussion Problems 53. yЉϪ2yЈϩ5y ϭ ex sin x 73. Suppose L is a linear differential operator that factors 1 but has variable coefficients. Do the factors of L com- 54. yЉϩyЈϩ y ϭ ex(sin 3x Ϫ cos 3x) 4 mute? Defend your answer. 4.6 VARIATION OF PARAMETERS REVIEW MATERIAL ● Basic integration formulas and techniques from calculus ● Review Section 2.3 INTRODUCTION We pointed out in the discussions in Sections 4.4 and 4.5 that the method of undetermined coefficients has two inherent weaknesses that limit its wider application to linear equations: The DE must have constant coefficients and the input function g(x) must be of the type listed in Table 4.4.1. In this section we examine a method for determining a particular solution yp of a nonhomogeneous linear DE that has, in theory, no such restrictions on it. This method, due to the eminent astronomer and mathematician Joseph Louis Lagrange (1736–1813), is known as varia- tion of parameters. Before examining this powerful method for higher-order equations we revisit the solution of lin- ear first-order differential equations that have been put into standard form. The discussion under the first heading in this section is optional and is intended to motivate the main discussion of this section that starts under the second heading. If pressed for time this motivational material could be assigned for reading. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4.6 VARIATION OF PARAMETERS ● 157 Linear First-Order DEs Revisited In Section 2.3 we saw that the general so- Јϩ ϭ lution of a linear first-order differential equation a1(x)y a0(x)y g(x) can be found by first rewriting it in the standard form dy ϩ P(x)y ϭ f(x) (1) dx and assuming that P(x) and f(x)are continuous on an common interval I. Using the in- tegrating factor method, the general solution of (1) on the interval I, was found to be ͵ ͵ ͵ ϭ Ϫ P(x)dx ϩ Ϫ P(x)dx͵ P(x)dx See (4) of Section 2.3. ᭤ y c1e e e f(x) dx. The foregoing solution has the same͵ form as that given in Theorem 4.1.6, namely, ϭ ϩ ϭ Ϫ P(x)dx y yc yp. In this case yc c1e is a solution of the associated homogeneous equation dy ϩ P(x)y ϭ 0 (2) dx ͵ ͵ ϭ Ϫ P(x)dx͵ P(x)dx and yp e e f (x) dx (3) is a particular solution of the nonhomogeneous equation (1). As a means of moti- vating a method for solving nonhomogeneous linear equations of higher-order we The basic procedure is ᭤ that used in Section 4.2. propose to rederive the particular solution (3) by a method known as variation of parameters. Suppose that y1 is a known solution of the homogeneous equation (2), that is, dy 1 ϩ P(x)y ϭ 0. (4) dx 1 ͵ ϭ Ϫ P(x)dx It is easily shown that y1 e is a solution of (4) and because the equation is linear, c1y1(x) is its general solution. Variation of parameters consists of finding a par- ϭ ticular solution of (1) of the form yp u1(x)y1(x). In other words, we have replaced the parameter c1 by a function u1. ϭ Substituting yp u1y1 into (1) and using the Product Rule gives d u y ϩ P(x)u y ϭ f(x) dx [ 1 1] 1 1 dy du u 1 ϩ y 1 ϩ P(x)u y ϭ f(x) 1 dx 1 dx 1 1 04 because of (4) dy du u 1 ϩ P(x)y ΅ ϩ y 1 ϭ f (x) 1 dx 1 1 dx du so y 1 ϭ f (x). 1 dx By separating variables and integrating, we find u1: ϭ f(x) ϭ ͵ f (x) du1 dx yields u1 dx. y1(x) y1(x) Hence the sought-after particular solution is ϭ ϭ ͵ f(x) yp u1y1 y1 dx. y1(x) ͵ ϭ Ϫ P(x)dx From the fact that y1 e we see the last result is identical to (3). Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 158 ● CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS Linear Second-Order DEs Next we consider the case of a linear second- order equation Љϩ Јϩ ϭ a2(x)y a1(x)y a0(x)y g(x), (5) although, as we shall see, variation of parameters extends to higher-order equations. The method again begins by putting (5) into the standard form yЉϩP(x)yЈϩQ(x)y ϭ f(x) (6) by dividing by the leading coefficienta2(x). In (6) we suppose that coefficient func- tions P(x), Q(x), and f(x) are continuous on some common interval I. As we have already seen in Section 4.3, there is no difficulty in obtaining the complementary ϭ ϩ solution yc c1y1(x) c2y2(x), the general solution of the associated homogeneous equation of (6), when the coefficients are constants. Analogous to the preceding dis- cussion, we now ask: Can the parameters c1 and c2 in yc can be replaced with func- tions u1 and u2, or “variable parameters,” so that ϭ ϩ y u1(x)y1(x) u2(x)y2(x) (7) is a particular solution of (6)? To answer this question we substitute (7) into (6). Using the Product Rule to differentiate yp twice, we get Ј ϭ Ј ϩ Ј ϩ Ј ϩ Ј yp u1y1 y1u1 u2y2 y2u2 Љ ϭ Љ ϩ Ј Ј ϩ Љ ϩ Ј Ј ϩ Љ ϩ Ј Ј ϩ Љ ϩ Ј Ј y p u1y1 y1u1 y1u1 u1y1 u2 y 2 y2u2 y2u 2 u2 y2. Substituting (7) and the foregoing derivatives into (6) and grouping terms yields 44zero zero Љ ϩ Ј ϩ ϭ Љ ϩ Ј ϩ ϩ Љ ϩ Ј ϩ ϩ Љ ϩ Ј Ј yp P(x)yp Q(x)yp u1[y 1 Py1 Qy1] u2[y2 Py2 Qy2] y1u 1 u1y1 ϩ Љ ϩ Ј Ј ϩ Ј ϩ Ј ϩ Ј Ј ϩ Ј Ј y2u2 u2 y2 P[y1u1 y2u2] y1u1 y2u2 d d ϭ [y uЈ] ϩ [y uЈ] ϩ P[y uЈ ϩ y uЈ] ϩ yЈuЈ ϩ yЈuЈ dx 1 1 dx 2 2 1 1 2 2 1 1 2 2 d ϭ [y uЈ ϩ y uЈ] ϩ P[y uЈ ϩ y uЈ] ϩ yЈuЈ ϩ yЈuЈ ϭ f (x).
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