A cable hanging under its own weight has slope … that satisfies…

A cable hanging under its own weight has slope $S=dy/dx$ that satisfies $dS/dx=c\sqrt{1+S^2}$. The constant $c$ is the ratio of cable density to tension.
(a) Show that $S=\sinh cx$ satisfies the equation.
(b) Integrate $dy/dx=\sinh cx$ to find the cable height $y(x)$, if $y(0)=1/c$.
(c) Sketch the cable hanging between $x=-L$ and $x=L$ and find how far it sags down at $x=0$.

Its source prints no answer for this problem, so there's nothing to check against.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §6.7 Hyperbolic Functions, #55