Examples of functions whose arc-length from the origin is given by their derivative

Question

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)

Answer 1

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