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 .
Attempt:
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?
I think it would be easier if you used Duhamel's principle. Assuming you know how to solve the homogenous case.