Book recommendations for Fourier Series, Dirac Delta Function and Differential Equations?

For a down to earth but rigorous account distributions and delta functions (but not so much differential equations) you can't beat James Lighthill's Introduction to Fourier analysis and generalised functions, Cambridge University Press. ISBN 978-0-521-05556-7.

The book is quite thin, only 70 pages or so. It is written at the undergraduate level. Although he uses test function (He calls them "good functions") to define how distributions work --- just as in advanced books for mathematicians--- there is not much sophisticated mathematical formalism and what there is, is well matched to what physics students learn.

The book has many applications to Fourier series and Fourier integrals of exactly the type one meets in physics papers and that are not often explained in regular mathematical methods classes. Lighthill is a great expositor (I took his "Waves in fluids" class when I was an undergrad and it was one of the best classes I had) and the book is well set out for self study. Amazon has used copies for about $14.

In my opinion the Schaum series offers a wide variety of questions and examples. They probably have different books for differential equations and Fourier series. They'll handle Dirac Delta in the differential equations one.