# Finding the correct try by method of undetermined coefficients

I'm trying an odd problem involving finding the proper form of the particular solution for several ODEs, using the method of undetermined coefficients.

The four equations are:

$$y'''-y''+4y'-4y=(x^2+11)e^x$$

$$y''-2y'+5y=(4x^2+2x)e^x$$

$$y''-11y'+18y=x^2e^{2x}$$

$$y''-2y'=1+xe^{-6x}$$

The aim is to find $y_p$ with the undetermined coefficients as P, Q, R, S etc. I tried out a couple, finding $Pe^x+Qxe^x +Rx^2e^x$ for equation 2 and $e^{2x}(Px^2+Qx+R)$ for 3 to no avail. Any help is appreciated, thank you.

• I do not believe the first has a solution (is it written correctly)? For the second, let $y_p = a e^x + b x e^x + c x^2 e^x$. You will get $a = -\dfrac{1}{2}, b = \dfrac{1}{2}, c = 1$. You should work the other two, which both have solutions too. – Amzoti Mar 2 '15 at 23:07
• @Amzoti You're correct, I wrote the question wrong, will edit immediately – jofl Mar 2 '15 at 23:16
• That is better. For the first, the homogeneous solution has an $e^x$ term, so choose $y_p = x(a e^x + b x e^x + c x^2 e^x)$. Hopefully now, you can work the other two. – Amzoti Mar 2 '15 at 23:21