I apologize in advance because I don't know how to type math, so I had to upload an image of my work.
The problem is:$4\frac{d^2y}{dx^2}+11\frac{dy}{dx}-3y=-2t e^{-3t}$. Find the general solution to that.
My work is attached here , and I don't understand why I'm not getting the correct solution (I'm getting close). I would greatly appreciate it if anyone could point out where my mistake(s) is/are. Thank you so much.
Your assumption about of a particular solution is wrong, since $y_p=Ate^{-3t}+Be^{-3t}$ contains a term which is a solution of the homogeneous associated DE, namely $Be^{-3t}$. And $y_H=c_1e^{-3t}+c_2e^{t/4}$.
So, try with $y_p=At^2e^{-3t}+Bte^{-t}$ which is the right form for a particular solution of the DE.