# Function which verifies certain conditions

I came upon an interesting equation and I have found no way to solve it yet:

$\int_{0}^{h} f(x) dx = f(h)*k$, where k is a real constant, h real and greater than 0, and f is a real-valued function with $f(0)=0$.

I have to find the function $f$. Any help will be appreciatted. Thank you.

## 1 Answer

Differentiating both sides with respect to $h$ and applying the fundamental theorem of calculus, we obtain $$f(h) = kf'(h)$$ This is a separable differential equation, and so its solution is given by $$\int \frac{k}{f}df = \int dh$$ $$k\ln(f) = h+c$$ $$f(h) = ce^{h/k}$$ Since $f(0) = 0$, we have $c = 0$ and so $f(h) = 0$ for all $h$. This is the only solution.

• I got to this kind of solution before asking the question but I had doubts about it. Thank you a lot for finding the time to clarify this for me. – tzuica Aug 15 '16 at 17:03