In Figure $11.3,$ the height, $y,$ of the hanging cable above the horizontal line satisfies
\[
\frac{d^{2} y}{d x^{2}}=k \sqrt{1+\left(\frac{d y}{d x}\right)^{2}}
\]
(a) Show that $y=\frac{e^{x}+e^{-x}}{2}$ satisfies this differential equation if $k=1$
(b) For general $k$, one solution to this differential equation is of the form
\[
y=\frac{e^{A x}+e^{-A x}}{2 A}
\]
Substitute this expression for $y$ into the differential equation to find $A$ in terms of $k$.