\[ y''(x)=y(x) \left (-\text {csch}^2(x)\right ) \left (-a^2 \sinh ^2(x)-(n-1) n\right ) \] ✓ Mathematica : cpu = 0.78002 (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 = 2.059 (sec), leaf count = 97
\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( \sinh \left ( x \right ) \right ) ^{n}{\mbox {$_2$F$_1$}(-{\frac {a}{2}}+{\frac {n}{2}},{\frac {a}{2}}+{\frac {n}{2}};\,{\frac {1}{2}};\,{\frac {\cosh \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}+{{\it \_C2}\, \left ( \sinh \left ( x \right ) \right ) ^{n} \left ( 2\,\cosh \left ( 2\,x \right ) +2 \right ) ^{{\frac {3}{4}}}{\mbox {$_2$F$_1$}({\frac {1}{2}}-{\frac {a}{2}}+{\frac {n}{2}},{\frac {1}{2}}+{\frac {a}{2}}+{\frac {n}{2}};\,{\frac {3}{2}};\,{\frac {\cosh \left ( 2\,x \right ) }{2}}+{\frac {1}{2}})}\sqrt [4]{2\,\cosh \left ( 2\,x \right ) -2}{\frac {1}{\sqrt {\sinh \left ( 2\,x \right ) }}}} \right \} \]