I just started reading Partial Differencial Equations. What you said is the material on Integral/Fourier Transforms.

I did not learn this yet but I did find it in my book.

Theorem:Given the heat equation withno boundary conditionsi.e. it needs to be satisfied on with (some well-behaved function). Then, the solution can be written as the Fourier Integral:

Now in your problem if:

In this case the solution is given by:

Which is the Gaussian Curve, and it is very similar to how you written it.