hw8solns - Section 4.5 1. y + y = e-x + x2 ; homogeneous...

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Section 4.5 1. y 00 + y = e - x + x 2 ; homogeneous solution r 2 + 1 = 0 r = ± i ; fundamental set is { cos x, sin x } . Particular UC sets: { e x } , { x 2 ,x, 1 } , so no modification is necessary. y p = Ae - x + Bx 2 + Cx + E 3. y 00 + y = 4 sin x + e x cos x ; homogeneous solution r 2 + 1 = 0; linearly independent solutions { cos x, sin x } , y c = C 1 cos x + C 2 sin x . Particular solution: 4 sin x ⇒ { sin x, cos x } ⇒ modify as { x cos x,x sin x } e x cos x ⇒ { e x cos x,e x sin x } ⇒ no modification necessary Hence y p = A 1 x cos x + A 2 x sin x + B 1 e x cos x + B 2 e x sin x 6. y 00 - y = e x + xe - x ; homogeneous solution r 2 - 1 = 0 r = ± 1; fundamental set is { e x ,e - x } . Particular solution: UC sets are { e x } , { xe - x ,e - x } , both need modification, to { xe x } , { x 2 e - x ,xe - x } . Thus y p = Axe x + Bx 2 e - x + Cxe - x 9. y 00 + 4 y 0 + 13 y = e - 2 x . We have y p = Ae - 2 x y 0 p = - 2 Ae - 2 x ,y 00 p = 4 Ae - 2 x y 00 p + 4 y 0 p + 13 yp = (4 Ae - 2 x
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This note was uploaded on 03/11/2008 for the course EC 201 taught by Professor Brown during the Winter '07 term at Cal Poly Pomona.

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hw8solns - Section 4.5 1. y + y = e-x + x2 ; homogeneous...

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