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When dealing with any mathematical problem, often the best way to do it is to break it up into smaller parts that are easier to deal with. So starting with our original linear inhomogeneous IVP,
We consider the simpler problems
(And, in order for uniqueness, all problems are coupled with the additional physical constraint that the integral of
If we have found a solution
What is nice about this approach is that (1) and (2) are already well studied problems. For the first, we use the fundamental solution of the heat equation, also known as the heat kernel, and for the second, we use the Green's function (see item 14 in the linked table).
In the following, assume
The fundamental solution is the solution
This is a very well known problem, and the solution is
The Green's function is the solution
And again, the solution of this problem is also well known,
Now that these two solutions are known, we can write
In your case with
Though often this procedure will not give you closed forms all that easily, it is the best way to deal with these kinds of problems in general, because typically closed forms are just not possible. To get closed forms, one would usually use separation of variables, but I stress that, in general, techniques like separation of variables will just fail.