First look for the fundamental solutions by solving the homogeneous version of the ODE:

The characteristic equation is

with roots  and
 and  , giving the two solutions
, giving the two solutions  and
 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.)
 anyway.)
Substitute these into the ODE:




 is already accounted for, so assume an ansatz of the form
 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
