\[ y''(x)=y(x) \left (-\csc ^2(x)\right ) \sec ^2(x) \left (-a \sin ^2(x) \cos ^2(x)-(m-1) m \sin ^2(x)-(n-1) n \cos ^2(x)\right ) \] ✓ Mathematica : cpu = 0.847682 (sec), leaf count = 615
\[\left \{\left \{y(x)\to \frac {c_2 (-1)^{\frac {1}{2} (-2 m-1)+1} \cos ^2(x)^{\frac {1}{4} (-2 m-1)+1} \left (\cos ^2(x)-1\right )^{\frac {1}{2} \left (\frac {4 a m+4 \sqrt {-a} n^2+4 a n-4 \sqrt {-a} n+4 (-a)^{3/2}+8 \sqrt {-a} a+\sqrt {-a}+4 m n^2-4 m n+m+4 n^3-4 n^2+n}{8 a+8 n^2-8 n+2}+\frac {1}{2} \left (-\sqrt {-a}+m+n\right )+\frac {1}{2} (-2 m-1)+1\right )-\frac {1}{4}} \, _2F_1\left (\frac {1}{2} (-2 m-1)+\frac {1}{2} \left (m+n-\sqrt {-a}\right )+1,\frac {1}{2} (-2 m-1)+\frac {4 n^3+4 m n^2+4 \sqrt {-a} n^2-4 n^2+4 a n-4 m n-4 \sqrt {-a} n+n+4 (-a)^{3/2}+8 \sqrt {-a} a+4 a m+m+\sqrt {-a}}{8 n^2-8 n+8 a+2}+1;\frac {1}{2} (-2 m-1)+2;\cos ^2(x)\right )}{\sqrt {\cos (x)}}+\frac {c_1 \cos ^2(x)^{\frac {1}{4} (2 m+1)} \left (\cos ^2(x)-1\right )^{\frac {1}{2} \left (\frac {4 a m+4 \sqrt {-a} n^2+4 a n-4 \sqrt {-a} n+4 (-a)^{3/2}+8 \sqrt {-a} a+\sqrt {-a}+4 m n^2-4 m n+m+4 n^3-4 n^2+n}{8 a+8 n^2-8 n+2}+\frac {1}{2} \left (-\sqrt {-a}+m+n\right )+\frac {1}{2} (-2 m-1)+1\right )-\frac {1}{4}} \, _2F_1\left (\frac {1}{2} \left (m+n-\sqrt {-a}\right ),\frac {4 n^3+4 m n^2+4 \sqrt {-a} n^2-4 n^2+4 a n-4 m n-4 \sqrt {-a} n+n+4 (-a)^{3/2}+8 \sqrt {-a} a+4 a m+m+\sqrt {-a}}{8 n^2-8 n+8 a+2};\frac {1}{2} (2 m+1);\cos ^2(x)\right )}{\sqrt {\cos (x)}}\right \}\right \}\] ✓ Maple : cpu = 0.966 (sec), leaf count = 102
\[ \left \{ y \left ( x \right ) = \left ( \sin \left ( x \right ) \right ) ^{n} \left ( \left ( \cos \left ( x \right ) \right ) ^{-m+1}{\mbox {$_2$F$_1$}({\frac {n}{2}}-{\frac {m}{2}}+{\frac {i}{2}}\sqrt {a}+{\frac {1}{2}},{\frac {n}{2}}-{\frac {m}{2}}-{\frac {i}{2}}\sqrt {a}+{\frac {1}{2}};\,{\frac {3}{2}}-m;\, \left ( \cos \left ( x \right ) \right ) ^{2})}{\it \_C2}+ \left ( \cos \left ( x \right ) \right ) ^{m}{\mbox {$_2$F$_1$}({\frac {n}{2}}+{\frac {m}{2}}+{\frac {i}{2}}\sqrt {a},{\frac {n}{2}}+{\frac {m}{2}}-{\frac {i}{2}}\sqrt {a};\,m+{\frac {1}{2}};\, \left ( \cos \left ( x \right ) \right ) ^{2})}{\it \_C1} \right ) \right \} \]