Non-homogeneous general solution = homogeneous general solution+non-homogeneous special solution

According to what you said, a special solution of the original equation in the form of y''+ay'+by = ce x is that y = e 2x+(1+x) e x is substituted into the differential equation.

Get A=-3 b=2 c=- 1, so the original equation becomes y''-3y'+2y =-e x, and the homogeneous solution is y = b 1e x+b2e 2x.

The general solution of the original equation is y = b 1e x+b2e 2x (homogeneous general solution) +e 2x+( 1+x) e x (inhomogeneous special solution).

Y = c 1e x+c2e 2x+xe x where c1= b1+1C2 = B2+1.

I hope this helps.