How to solve an equation that contains a function. I was trying to solve a question . And I got this equation (the function is continuous):
$$x .cos(\pi.x) = f(x^2)-f(0)$$
I wanted to use Maple to find what $f(x)$ is but couldn't. I even don't know whether it is possible to find a function providing this equation. Actually I'm trying to find the value of $f'(4)$, but I'd be appreciated if there's a way to solve this kind of problems using Maple.
Edit: I asked a bad question, because I chose a wrong problem. Actually, I was trying to ask a general question. In another question, I have this information: $y = f(x)$ is continuos. And I have the graph of $y=f(x)$. So when I reach here (an equation that contains differentials and integrals),
$$\int_{1}^{5} \frac{f(x)-x.f'(x)}{f^2(x)} dx = c $$
If $c$ is a constant. How can I find $f(x)$, using this equation. Is it possible and can I use Maple for problems like these.
Kind of trivially, we have that
$$f(x) = \sqrt{x}\cos\left(\pi\sqrt{x}\right)+f(0)$$
But this is not necessary to find $f'(4)$. Take the derivative of both sides of that expression
$$\cos(\pi x) - \pi x \sin(\pi x) = 2xf'(x^2)$$
then plug in $x = 2$ to get
$$1 - 0 = 4f'(4) \implies f'(4) = \frac{1}{4}$$
technically the solution is not unique if you allow negative $x$, which is why I wrote the function assuming $x$ was always positive.