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Question

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?

My attempt

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.

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