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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 $C$.

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$x$ ranges over what, all reals including negative? Values of $h$ are what, integers? –  GEdgar Dec 1 '11 at 14:33
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1 Answer 1

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.

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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
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$h^2$ 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
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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
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