Question

d²y/dx²-6dy/dx+9y=x²e^(3x)

Answer 1

Given differential equation is $\frac{d^2y}{dx^2}-6\frac{dy}{dx}+9y=x^2e^{3x}$

Auxiliary equation is, $m^2-6m+9 = 0 \implies m =3,3$

C.F. $y = (c_1+c_2x)e^{3x}$

P.I. $\frac{1}{D^2-6D+9}x^2e^{3x} = e^{3x}\frac{1}{(D+3)^2-6(D+3)+9}x^2$

$=e^{3x}\frac{1}{D^2}x^2 = e^{3x}\int \int x^2dx dx = \frac{x^4}{12}e^{3x}$

Hence complete solution of the equation is,

$y = (c_1+c_2x)e^{3x}+\frac{x^4}{12}e^{3x}$