Question

(D²+2)y=x²e³x+excos2x

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Answer 1

(D^2 + 2D) y= x^2

It's Particular integral is

=> PI = [1/(D^2 +2D) ]x^2

=> PI = [1/2D(1+D/2) ]x^2

=> PI = [1/2D(1+D/2) ^-1]x^2

=> PI = [1/2D(1- D/2 + D^2/4- D^3/8)]x^2

=> PI = [1/2D- 1/4+ D/8 - D^2/16]x^2

PI = x^3/6 - x^2/4+x/4 -1/8.