Problem source

\begin{questionparts}
\item Solve the equation $u^2+2u\sinh x -1=0$ giving $u$ in terms
of $x$.
Find the solution of the differential equation
\[
\left( \frac{\d y}{\d x}\right)^{\!2} +2 \frac{\d y}{\d x} \sinh x -1 = 0
\]
that satisfies $y=0$ and $\dfrac {\d y}{\d x} >0$ at $x=0$.
\item
Find the solution, not identically zero,  of the differential equation 
\[
\sinh y  \left( \frac{\d y}{\d x}\right)^{\!2} 
+2 \frac{\d y}{\d x}  -\sinh y = 0
\]
that satisfies $y=0$ at $x=0$,
expressing your solution in the form
$\cosh  y=\f(x)$. Show that the asymptotes to the solution curve are
$y=\pm(-x+\ln 4)$.
\end{questionparts}