\(\displaystyle{u}{''}={\left({\sin{{\left({y}\right)}}}\right)}{''}={\left({\cos{{\left({y}\right)}}}{y}'\right)}'={\cos{{\left({y}\right)}}}{y}{''}-{\sin{{\left({y}\right)}}}{y}'^{{2}}={c}{\sin{{\left({y}\right)}}}={c}{u}\)

which is, depending on the sign of c, an oscillation equation or an exponential function which is easily solvable for u and thus for y.

\[u=\begin{cases}a_1\cos(\sqrt{|c|}x)+a_2\sin(\sqrt{|c|}x)&for\ c<0\\b_1+b_2x&for\ c=0\\c_1e^{\sqrt{c}x}+c_2e^{\sqrt{c}x}&for\ c>0\end{cases}\]