$$ y''+ 3y' - 4y = e^x + 5xe^{-4x} $$ What would the particular integral be?
I'm using the method of undetermined coefficients. I've tried to guess so my possible solutions but none of them satisfy the equation.
e.g. $y_p = xe^x + e^{-4x}(Ax + B)$ ---- this doesn't work.
Hint. Note that $1$ and $-4$ are solutions of the characteristic equation. Therefore you should try with $$y_p(x) = Cxe^x + xe^{-4x}(Ax + B).$$ After the substitution we find that for all $x\in\mathbb{R}$, $$5Ce^x+(2A-5B-10Ax)e^{-4x}=e^x + 5xe^{-4x}$$ which holds iff $$5C=1,\quad -10A=5,\quad 2A-5B=0.$$