Using exponential generating functions to solve recurrence equations

Question

It needs to solve $a_{n+1}=3a_{n}+5$ by using an exponential generating function. I tried to solve it and finaly got next equation: $E'(t)=3E(t)+5e^t$. I do not know what to do next, may be I choose not right way of solution. Thank you in advance.

Answer 1

Well solve the ODE !

The solution of the homogeneous equation $\;E'(t)-3E(t)=0\;$ is $\;E(t)=A\,e^{3t}\,$ and using (for example) variation of constant $A:=A(t)\,$ you may deduce that $A'(t)=5\,e^{-2t}$ and conclude.