I was wondering about the following:
Let's say there is a differential equation whose solution is $f$
And $f$ also satisfies a functional equation.
Can anyone construct an (non-trivial) example where knowing the functional equation gives some sort of advantage in solving the differential equation or visa-versa? And if the functional equation does not help can you please give your reasoning on why so?
For example, take $$df/dx = f$$ And the functional equation is of the form: $$ A f(x+y) = f(x)f(y)$$ where, A is a constant. I can't see any sort manipulation where knowing the functional equation has given be an advantage in solving the differential equation.