why not make it generally at once?
consider by knowing that is a solution of the homogeneous equation, then substitute and prove that
is a solution to do related homogeneous equation, and I need to find a second linearly independent solution of the DE using reduction of order.
So,
Plugging into the homogeneous DE:
After some simplification, I get
I know that for reduction of order to work, the terms involving V need to go away, but I can't see a way to make that happen.