First look for the fundamental solutions by solving the homogeneous version of the ODE:
The characteristic equation is
with roots and , giving the two solutions and .
For the non-homogeneous version, you can exploit the superposition principle and consider one term from the right side at a time.
Assume the ansatz solution,
(You could include a constant term <em>f</em> here, but it would get absorbed by the first solution anyway.)
Substitute these into the ODE:
is already accounted for, so assume an ansatz of the form
Substitute into the ODE:
Assume an ansatz solution
Substitute into the ODE:
So, the general solution of the original ODE is