Answer
$$y'' = {a^2}y$$
Work Step by Step
$$\eqalign{
& {\text{Let }}y = \sinh \left( {ax} \right) \cr
& {\text{Calculate }}y'{\text{ and }}y'' \cr
& y' = \frac{d}{{dx}}\left[ {\sinh \left( {ax} \right)} \right] \cr
& y' = \cosh \left( {ax} \right)\left( a \right) \cr
& y' = a\cosh \left( {ax} \right) \cr
& and \cr
& y'' = {a^2}\sinh \left( {ax} \right) \cr
& y'' = {a^2}\underbrace {\sinh \left( {ax} \right)}_y \cr
& {\text{Then}} \cr
& y'' = {a^2}y \cr} $$
