Answer to Question #199888 in Differential Equations for Ajay

Question

(D² - 2DD' + D'²)z = x²y²ex+y

Answer 1

$(D-D')^2z=0$

General solution:

C.F+P.I

C.F.:

$\phi_1(y+x)+x\phi_2(y+x)$

P.I.:

$\frac{1}{(D-D')(D-D')}(x²y²e^x+y)$

$\frac{1}{D-D'}(x²y²e^x+y)=\int(x²y²e^x+y)dx=y^2(x^2-2x+2)e^x+xy$

$\frac{1}{D-D'}(y^2(x^2-2x+2)e^x+xy)=\int(y^2(x^2-2x+2)e^x+xy)dx=$

$=y^2(x^2-4x+6)e^x+x^2y/2$