5
$\begingroup$

I am stuck on the problem:

Find all continuous functions $h$ satisfying $\int_{0}^{x}h(y)dy=\left [ h(x) \right ]^{2}+C$ for some constant $C$.

  • 0
    x ranges over what, all reals including negative? Values of $h$ are what, integers?2011-12-01

1 Answers 1

2

HINT: The left hand side is known to be differentiable by the fundamental theorem of calculus, so the right hand side is also differentiable. Differentiate both sides to form a differential equation, and then solve that.

  • 1
    Since $h(x)$ is continuous, it is differentiable on any interval where $h(x) \neq 0$ since $h(x) = \pm \sqrt{(h(x))^2}$ on that interval. So you can solve as suggested here on any such interval, and then when you're done see if there's any issue with "matching" up the different solutions at points $x$ where $h(x) = 0$. Here you can have a function like $h(x) = {x \over 2}$ for x > 0 and $h(x) = 0$ otherwise.2011-12-01