2023 THREE(e)

A power line hangs between two poles.

The equation of the curve $y=f(x)$ that models the shape of the
power line can be found by solving the differential equation:

$a\frac{d^{2}y}{d{x}^2}=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$

Use differentiation to verify that the function $y=\frac{a}{2}\left(e^{\frac{x}{a}}+e^{-\frac{x}{a}}\right)$

satisfies the above differential equation, where $a$ is a positive
constant.