its a bit tedeous...

firstly we have to consider the homogenous equation

which has solution

using the variation of parameters theorem we see that

with calculted using the Wronskian determinant

so we can then see that

and after substituting and turning the crank we end up with

which is simple to integrate, we then find the final answer of

The final anwer will then just be the superposition of the particular and complimentary solutions