Fourier transform to solve inhomogeneous PDE

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?

Re: Fourier transform to solve inhomogeneous PDE

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

Re: Fourier transform to solve inhomogeneous PDE

Quote:

Originally Posted by

**Jose27** I think it would be easier if you used

Duhamel's principle. Assuming you know how to solve the homogenous case.

Thank you. But I have not studied Duhamel's principle before and it is not in my lecture notes. There must be another way to solve the problem.