Functional equations are some of my favorite types of math puzzles. I was wondering if anybody knows of any good (and solvable) functional equation problems that are combined with a differential equation. For example (though I know not the solutions or if it has any): $$f(2x)+f''(x+1)=\sqrt{1+f'(x)}$$ Can anybody give any good examples of solvable combination functional/differential equation problems and perhaps also give me some ideas about how to solve them using non-trivial solutions (constant functions)?
Thanks!