We have found the following particular solution and its derivatives. Yp = Ax² + Bx + C + (Dx + E)e* 7 Yp = 2AX + B + (Dx + E)ex + De* Yp = 2A + (Dx + E)e* + 2De* Substituting into the original differential equation results in the following. y" - 8y' + 20y = 200x² - 52xe* (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 + De*) +20(Ax2 + Bx + C + (Dx + E)e*) = (2A8B + 20C) + (-16A + 20B)x+ (-6D+ 13E)e + -( D = -52 20(Ax2 + Bx + C + (Dx + E)ex) Joxe*+( AX² As the coefficients of the terms in this simplified expression must be equal to the coefficients of 200x2 - 52xe, we have the following system 2A8B+20C = 0 -16A + 20B = 0 -6D+ 13E = 0 A = 200 = 200x² - 52xe*
We have found the following particular solution and its derivatives. Yp = Ax² + Bx + C + (Dx + E)e* 7 Yp = 2AX + B + (Dx + E)ex + De* Yp = 2A + (Dx + E)e* + 2De* Substituting into the original differential equation results in the following. y" - 8y' + 20y = 200x² - 52xe* (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 + De*) +20(Ax2 + Bx + C + (Dx + E)e*) = (2A8B + 20C) + (-16A + 20B)x+ (-6D+ 13E)e + -( D = -52 20(Ax2 + Bx + C + (Dx + E)ex) Joxe*+( AX² As the coefficients of the terms in this simplified expression must be equal to the coefficients of 200x2 - 52xe, we have the following system 2A8B+20C = 0 -16A + 20B = 0 -6D+ 13E = 0 A = 200 = 200x² - 52xe*
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:We have found the following particular solution and its derivatives.
Yp = Ax² + Bx + C + (Dx + E)e*
2AX + B + (Dx + E)ex + Dex
17
Yp
= 2A + (Dx + E)e* + 2De*
Substituting into the original differential equation results in the following.
y" - 8y' + 20y = 200x² - 52xe*
YP,
(2A + (Dx + E)ex + 2Dex) - 8(2AX + B + (Dx + E)ex + Dex) + 20(Ax² + Bx + C + (DX + E)ex) = 200x² - 52xe*
Simplifying the left side of this equation gives the following.
(2A + (Dx + E)ex + 2Dex) - 8(2Ax + B + (Dx + E)ex + Dex)
+ 20(Ax² + Bx + C + (Dx + E)e*)
=
= (2A - 8B + 20C) + (−16A + 20B)x + (−6D + 13E)ex +
(
Joxe* + (
JAx²
As the coefficients of the terms in this simplified expression must be equal to the coefficients of 200x2 - 52xe, we have the following system
2A8B20C = 0
-16A + 20B = 0
-6D + 13E = 0
- -52
A =
= 200
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