Substituting and its derivatives into the ODE gives
as required.
Assume a second solution of the form . Then
Substitute these into the ODE:
This ODE is separable as
Integrating both sides gives
For the remaining integral, we have
So we have
Integrate both sides to solve for :
Then our second solution turns out to be
already captures the second term here, so the second fundamental solution is
and the general solution is