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 $$\frac {\mathrm d f}{\mathrm d x} = f$$ 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.
You can transform functional equations into differential equations and into difference equations. For example, if $f$ satisfies $$f(x+y)=f(x)f(y)$$ then it must satisfy $f(x+1)-f(x)=f(x)(f(1)-1)$ or $$\Delta f=Cf.$$ This difference equation is trivial and has the solution $f=f(1)^x$. Likewise, the function $f$ in
$$f(x+y)=yf^2(x)+(y^2+1)f(x)$$ must solve (by putting $y=dx$)
$$\frac{f(x+dx)-f(x)}{dx}=f'(x)=f^2(x).$$ Or it must solve $$\Delta f =f^2+f.$$ Just make sure to match the appropriate constants from the functional equation to the difference or differential equation. A differential equation might not always be derivable from a functional equation, but a difference equation is guaranteed. Most of the time you will need series solutions.