I was wondering if the equation $\frac{d^2x}{dt^2} = -a\sinh(x)$ had an exact solution because the differential equation $\frac{d^2x}{dt^2} = -a\sin(x)$ has an exact solution by way of the elliptical integral.
This leads me to believe that, since $\sinh(x) = -i\sin(ix)$, $\frac{d^2x}{dt^2} = -a\sinh(x)$ should have a similar solution. Is this so?
Yes: If $x = f(t)$ satisfies $f''(t) = -a\sin[f(t)]$, then $y = -i\, f(t)$ satisfies $$ \frac{d^{2}y}{dt^{2}} = -if''(t) = ia\sin[f(t)] = ia\sin(iy) = -a\sinh(y). $$