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We have found the following particular solution and its derivatives. = Ax² + Bx + C + (DX + E)ex Yp = 2AX + B + (Dx + E)ex + Dex = 2A + (DX + E)ex + 2Dex "p Substituting into the original differential equation results in the following. y" - 8y' + 20y = 200x278xex (2A + (DX + E)ex + 2Dex) − 8(2Ax + B + (Dx + E)ex + Dex) + 20(Ax² + Bx + C + (Dx + E)ex) = 200x² - 78xex Simplifying the left side of this equation gives the following. (2A + (Dx + E)ex + 2Dex) - 8(2Ax + B + (Dx + E)e* + Dex) + 20(Ax² + Bx + C + (Dx + E)ex) = (2A - 8B + 20C) + (-16A + 20B)x+ (-6D + 13E)ex + AX² As the coefficients of the terms in this simplified expression must be equal to the coefficients of 200x278xex, we have the following system. 2A8B20C = 0 -16A + 20B = 0 -6D + 13E = 0 = -78 = 200 Dxex +

We have found the following particular solution and its derivatives. = Ax² + Bx + C + (DX + E)ex Yp = 2AX + B + (Dx + E)ex + Dex = 2A + (DX + E)ex + 2Dex "p Substituting into the original differential equation results in the following. y" - 8y' + 20y = 200x278xex (2A + (DX + E)ex + 2Dex) − 8(2Ax + B + (Dx + E)ex + Dex) + 20(Ax² + Bx + C + (Dx + E)ex) = 200x² - 78xex Simplifying the left side of this equation gives the following. (2A + (Dx + E)ex + 2Dex) - 8(2Ax + B + (Dx + E)e* + Dex) + 20(Ax² + Bx + C + (Dx + E)ex) = (2A - 8B + 20C) + (-16A + 20B)x+ (-6D + 13E)ex + AX² As the coefficients of the terms in this simplified expression must be equal to the coefficients of 200x278xex, we have the following system. 2A8B20C = 0 -16A + 20B = 0 -6D + 13E = 0 = -78 = 200 Dxex +

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 5E
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We have found the following particular solution and its derivatives. Ax² + Bx + C + (Dx + E)ex = 2AX + B + (Dx + E)ex + Dex = 2A + (Dx + E)ex + 2Dex Ур Yo P = Yp Substituting into the original differential equation results in the following. y" - 8y' + 20y = 200x² - 78xex (2A + (Dx + E)ex + 2Dex) − 8(2Ax + B + (Dx + E)ex + Dex) + Simplifying the left side of this equation gives the following. (2A + (Dx + E)ex + 2De*) − 8(2Ax + B + (Dx + E)ex + Dex) + 20(Ax² + Bx + C + (Dx + E)ex) = (2A - 8B + 20C) + (−16A + 20B)x + (−6D + 13E)ex + ])pxex + (1 AX² As the coefficients of the terms in this simplified expression must be equal to the coefficients of 200x² - 78xex, we have the following system. 2A8B20C = 0 -16A + 20B = 0 -6D + 13E = 0 = -78 20(Ax² + Bx + C + (Dx + E)ex) = 200x² - 78xe* A = 200
Transcribed Image Text:We have found the following particular solution and its derivatives. Ax² + Bx + C + (Dx + E)ex = 2AX + B + (Dx + E)ex + Dex = 2A + (Dx + E)ex + 2Dex Ур Yo P = Yp Substituting into the original differential equation results in the following. y" - 8y' + 20y = 200x² - 78xex (2A + (Dx + E)ex + 2Dex) − 8(2Ax + B + (Dx + E)ex + Dex) + Simplifying the left side of this equation gives the following. (2A + (Dx + E)ex + 2De*) − 8(2Ax + B + (Dx + E)ex + Dex) + 20(Ax² + Bx + C + (Dx + E)ex) = (2A - 8B + 20C) + (−16A + 20B)x + (−6D + 13E)ex + ])pxex + (1 AX² As the coefficients of the terms in this simplified expression must be equal to the coefficients of 200x² - 78xex, we have the following system. 2A8B20C = 0 -16A + 20B = 0 -6D + 13E = 0 = -78 20(Ax² + Bx + C + (Dx + E)ex) = 200x² - 78xe* A = 200
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