Solve $(D^2 + 3D + 2)y = e^{e^x}$ using method of variation of parameters? This question was asked in a test and I'm stuck while solving this using method of variation of parameters. Here's an screenshot of my solution,




While finding particular integral I was not able to solve integration of e^e^x?
Can anyone please help me out a bit here. I was not able to understand how to solve it further.
You are almost there, but note that $W(x)=e^{-3x}$
$$PI=-e^{-2x} \int e^{2x} e^{e^{x}} dx + e^{x} \int e^{x} e^{e^{x}} dx$$
Let $e^x =t \implies dx=\frac{dt}{t},$ then
$$PI=-e^{-2x} \int t e^{t} dt + e^{-x} \int e^{t} dt$$
$$PI=-e^{-2x}e^{t}(t-1)+ e^{-x} e^{t}= e^{-2x+e^{x}} .$$
the final answer should be
$PI = e^{-2x} * e^{e^x}$