Riemann-integration

Question

Let $f:[0,\infty )\rightarrow R$ a real valued continous function such that: $f(x)\neq 0 \quad \forall x>0 \quad$ and

${ (f(x)) }^{ 2 }=2\int _{ 0 }^{ x }{ f(t)dt } \quad \forall x\ge 0$.

Prove that $f(x)=x\quad \forall x\ge 0$.

Answer 1

$$f(x)f'(x) = f(x) ~\forall x \ge 0.$$

If $x > 0$ then $f'(x) = 1.$ So, $f(x) = x + c$, where $c$ is a constant.

If $x = 0$ then $f(0) = 0.$

So, $c = 0.$ And then, $f(x) = x.$