\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-{\frac { \left ( -{a}^{2} \left ( \sinh \left ( x \right ) \right ) ^{2}-n \left ( n-1 \right ) \right ) y \left ( x \right ) }{ \left ( \sinh \left ( x \right ) \right ) ^{2}}}=0} \]
Mathematica: cpu = 1.132144 (sec), leaf count = 231 \[ \left \{\left \{y(x)\to \frac {c_2 (-1)^{\frac {1}{2} (-2 n-1)+1} \tanh ^2(x)^{\frac {1}{4} (-2 n-1)+1} \left (\tanh ^2(x)-1\right )^{\frac {1}{2} \left (\frac {a+n}{2}+\frac {1}{2} (a+n+1)+\frac {1}{2} (-2 n-1)+1\right )-\frac {1}{2}} \, _2F_1\left (\frac {1}{2} (-2 n-1)+\frac {a+n}{2}+1,\frac {1}{2} (-2 n-1)+\frac {1}{2} (a+n+1)+1;\frac {1}{2} (-2 n-1)+2;\tanh ^2(x)\right )}{\sqrt {\tanh (x)}}+\frac {c_1 \tanh ^2(x)^{\frac {1}{4} (2 n+1)} \left (\tanh ^2(x)-1\right )^{\frac {1}{2} \left (\frac {a+n}{2}+\frac {1}{2} (a+n+1)+\frac {1}{2} (-2 n-1)+1\right )-\frac {1}{2}} \, _2F_1\left (\frac {a+n}{2},\frac {1}{2} (a+n+1);\frac {1}{2} (2 n+1);\tanh ^2(x)\right )}{\sqrt {\tanh (x)}}\right \}\right \} \]
Maple: cpu = 0.187 (sec), leaf count = 97 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( \sinh \left ( x \right ) \right ) ^{n} {\mbox {$_2$F$_1$}(-{\frac {a}{2}}+{\frac {n}{2}},{\frac {a}{2}}+{\frac {n}{2}};\,{\frac {1}{2}};\,{\frac {\cosh \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})} +{{\it \_C2}\, \left ( \sinh \left ( x \right ) \right ) ^{n} \left ( 2\, \cosh \left ( 2\,x \right ) +2 \right ) ^{{\frac {3}{4}}} {\mbox {$_2$F$_1$}({\frac {1}{2}}-{\frac {a}{2}}+{\frac {n}{2}},{\frac {1}{2}}+{\frac {a}{2}}+{\frac {n}{2}};\,{\frac {3}{2}};\,{\frac {\cosh \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})} \sqrt [4]{2\,\cosh \left ( 2\,x \right ) -2}{\frac {1}{\sqrt {\sinh \left ( 2\,x \right ) }}}} \right \} \]