Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $C$.

share|cite|improve this question
$x$ ranges over what, all reals including negative? Values of $h$ are what, integers? – GEdgar Dec 1 '11 at 14:33

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.

share|cite|improve this answer
I'm not sure how you'd carry out this computation. I agree that the RHS is differentiable, but the Chain Rule doesn't apply, so in principle you can't say much about $\big([h(x)]^2\big)^{\prime}$. Could you clarify? – student Dec 1 '11 at 13:29
$h^2$ is differentiable, bot $h$ may not be. – Julián Aguirre Dec 1 '11 at 13:30
Well, how about first solving with the assumption $h$ is differentiable, then worry about whether there are additional solutions. – GEdgar Dec 1 '11 at 14:34
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. – Zarrax Dec 1 '11 at 15:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.