I think it would be easier if you used Duhamel's principle. Assuming you know how to solve the homogenous case.
Express the solution to the inhomogeneous equation
satisfying the initial condition
as an integral involving the source term .
The given solution is .
Take Fourier transform with respect to .
Solve the ODE by integrating factor.
Find inverse transform of .
Have I got above correct? Why should I use definite integral instead of indefinite integral for solving ? What should I do next? (I know I am using instead of in the given solution but I don't see how this is even close to the solution).
Should I expand like below?