Let \( f : [0, \infty) \to [0, \infty) \) be a differentiable function with \( f(x)>0 \) for all \( x>0 \), and \( f(0) = 0 \). Further, \( f \) satisfies}
\[
(f(x))^2 = \int_{0}^{x} \left( (f(t))^2 + f(t) \right) \, dt, \, x>0.
\]
{Then which one of the following options is correct?