We have found the following particular solution and its derivatives. Yp = Ax² + Bx + C + (Dx + E)e* Yp = 2AX + B + (Dx + E)e* + Dex = 2A + (Dx + E)e* + 2De* "1 Ур Substituting into the original differential equation results in the following. y" - 8y' + 20y = 200x² - 52xe* (2A + (Dx + E)e* + 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 + De*) + 20(Ax² + Bx + C + (Dx + E)e*) = (2A - 8B + 20C) + (-16A + 20B)x+ (-6D+ 13E)e* + ²+( 13 ✓ )oxe* + (20 4x² 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 = -6D + 13E=0 x )D x 4 = 200 = -52

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