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)