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I'm looking for functions $y:\mathbb{R}\rightarrow\mathbb{R}$ such that

$$\int_{0}^{a} \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx = \frac{dy}{dx}\Bigg|_{a}$$

(this kind of feels like a calculus-of-variations type problem, but I don't have any experience with the calculus of variations)

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Looks like a differential equations problem to me. Differentiate both side with respect to $a$. (Makes me uncomfortable, $a$ is usually a constant. Why not in the integral have $\frac{dy}{dt}$, and upper limit of integration $x$?) – André Nicolas Feb 24 '12 at 1:44
up vote 6 down vote accepted

The solutions are $y(x) = A + \cosh(x)$ for arbitrary constants $A$.

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Excellent, thank you. – deoxygerbe Feb 24 '12 at 1:56
A catenary – Henry Feb 24 '12 at 2:16

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