Hi. I have to solve this:

y''-3y'+2y=e^x,
Using the replacement y=\phi y_1 being y_1 a solution of the homogeneous differential equation. I can't do it "traditionally", I have to use this method.

So I have to solve that \phi ''y_1+ \phi ' \left[2y_1'+Py_1]  \right]=e^x

So, this is what I did:

y_1=C_1 e^x+C_2e^{2x}\rightarrow y_1'=C_1 e^x+2C_2e^{2x}
Then:
\phi ''C_1 e^x+\phi '' C_2 e^{2x}+ \phi ' \left[C_2e^{2x}-C_1e^x  \right]=e^x
What should I do from here? I don't know how to handle \phi

Bye there.