Hi, I'm doing 18.03SC and have run into a problem when getting the particular solution to the equation y^(4) - 2y^(2) + y = xe^x. My estimate of the particular solution is x^2(Ax+B)e^x since p(D) = (D^2-1)^2. I tried applying p(D) onto the estimate as such:
p(D)(e^x)(x^2)(Ax+B) = xe^x
(D^2-1)^2(e^x)(x^2)(Ax+ B) = xe^x
Using exponential shift
e^x * D^4 (x^2(Ax+B)) = xe^x. However this won't allow computation of A or B. Am I missing something here?

Question

Hi, I'm doing 18.03SC and have run into a problem when getting the particular solution to the equation y^(4) - 2y^(2) + y = xe^x. My estimate of the particular solution is x^2(Ax+B)e^x since p(D) = (D^2-1)^2. I tried applying p(D) onto the estimate as such: 
p(D)(e^x)(x^2)(Ax+B) = xe^x
(D^2-1)^2(e^x)(x^2)(Ax+ B) = xe^x
Using exponential shift
e^x * D^4 (x^2(Ax+B)) = xe^x. However this won't allow computation of A or B. Am I missing something here?