Find the exponential generating function of the sequence $\{na_n\}_{n\in\Bbb N}$ where $\{a_n\}_{n\in\Bbb N}$ is given by $a_0=-2,a_n=4a_{n-1}+3^n,n\ge 1.$
My attempt:
Let $f(x)=\displaystyle\sum_{n=0}^\infty\frac{a_n}{n!}x^n.$ Then: $$\begin{aligned}\sum_{n=0}^\infty\frac{na_n}{n!}x^n=xf'(x)&=\sum_{n=1}^\infty\frac{4a_{n-1}+3^n}{(n-1)!}x^n\\&=4x\sum_{n=1}^\infty\frac{a_{n-1}}{(n-1)!}x^{n-1}+3x\sum_{n=1}^\infty\frac{(3x)^{n-1}}{(n-1)!}\\&=4x\sum_{n=0}^\infty\frac{a_n}{n!}x^n+3x\sum_{n=0}^\infty\frac{(3x)^n}{n!}\\&=4xf(x)+3xe^{3x}\\\implies xf'(x)&=4xf(x)+3xe^{3x}\\\implies f'(x)-4f(x)&=3e^{3x}.\end{aligned}$$
Now, I got stuck because I don't know how to solve this differential equation. How can I solve this? Should I take a different route?