\[ \boxed { \left ( {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) \right ) y \left ( x \right ) - \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+ \left ( \tan \left ( x \right ) +\cot \left ( x \right ) \right ) y \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-{n}^{2} \left ( \cot \left ( x \right ) \right ) ^{2} \right ) \left ( y \left ( x \right ) \right ) ^{2}\ln \left ( y \left ( x \right ) \right ) =0} \]
Mathematica: cpu = 635.571707 (sec), leaf count = 915 \[ \left \{\left \{y(x)\to e^{e^{c_2+\int _1^x -\frac {\frac {(-1)^{1-n} 2^{n+1} \left (\frac {1}{2} (-n-1)+\frac {1}{2}\right ) K_n\left (\sqrt {\cos ^2(K[2])-1}\right ) \cos (K[2]) \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}} \sin (K[2]) \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)-\frac {1}{2}}}{\sqrt {1-\cos ^2(K[2])}}+\frac {(-1)^{1-n} 4^{\frac {n+1}{2}-\frac {1}{2}} K_n\left (\sqrt {\cos ^2(K[2])-1}\right ) \cos (K[2]) \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}} \sin (K[2]) \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)+\frac {1}{2}}}{\left (1-\cos ^2(K[2])\right )^{3/2}}+\frac {(-1)^{1-n} 2^{n+1} (n+1) K_n\left (\sqrt {\cos ^2(K[2])-1}\right ) \cos (K[2]) \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}-1} \sin (K[2]) \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)+\frac {1}{2}}}{\sqrt {1-\cos ^2(K[2])}}+\frac {\left (-\frac {1}{2}\right )^{1-n} \left (-K_{n-1}\left (\sqrt {\cos ^2(K[2])-1}\right )-K_{n+1}\left (\sqrt {\cos ^2(K[2])-1}\right )\right ) \cos (K[2]) \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}} \sin (K[2]) \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)}}{\sqrt {1-\cos ^2(K[2])}}+c_1 \left (-\frac {2^{n+1} \left (\frac {1}{2} (-n-1)+\frac {1}{2}\right ) I_n\left (\sqrt {\cos ^2(K[2])-1}\right ) \cos (K[2]) \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}} \sin (K[2]) \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)-\frac {1}{2}}}{\sqrt {1-\cos ^2(K[2])}}-\frac {4^{\frac {n+1}{2}-\frac {1}{2}} I_n\left (\sqrt {\cos ^2(K[2])-1}\right ) \cos (K[2]) \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}} \sin (K[2]) \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)+\frac {1}{2}}}{\left (1-\cos ^2(K[2])\right )^{3/2}}-\frac {2^{n+1} (n+1) I_n\left (\sqrt {\cos ^2(K[2])-1}\right ) \cos (K[2]) \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}-1} \sin (K[2]) \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)+\frac {1}{2}}}{\sqrt {1-\cos ^2(K[2])}}-\frac {2^{n-1} \left (I_{n-1}\left (\sqrt {\cos ^2(K[2])-1}\right )+I_{n+1}\left (\sqrt {\cos ^2(K[2])-1}\right )\right ) \cos (K[2]) \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}} \sin (K[2]) \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)}}{\sqrt {1-\cos ^2(K[2])}}\right )}{\frac {(-1)^{1-n} 4^{\frac {n+1}{2}-\frac {1}{2}} K_n\left (\sqrt {\cos ^2(K[2])-1}\right ) \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)+\frac {1}{2}} \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}}}{\sqrt {1-\cos ^2(K[2])}}-\frac {4^{\frac {n+1}{2}-\frac {1}{2}} I_n\left (\sqrt {\cos ^2(K[2])-1}\right ) c_1 \left (\cos ^2(K[2])-1\right )^{\frac {1}{2} (-n-1)+\frac {1}{2}} \left (2 \cos ^2(K[2])-2\right )^{\frac {n+1}{2}}}{\sqrt {1-\cos ^2(K[2])}}} \, dK[2]}}\right \}\right \} \]
Maple: cpu = 0.390 (sec), leaf count = 81 \[ \left \{ y \left ( x \right ) ={1{{\rm e}^{{\frac {{{\sl Y}_{n}\left ( \sin \left ( x \right ) \right )}{\it \_C2}}{\sin \left ( x \right ) \left ( {{\sl J}_{n}\left (\sin \left ( x \right ) \right )}{{\sl Y}_{n+1 }\left (\sin \left ( x \right ) \right )}-{{\sl J}_{n+1}\left (\sin \left ( x \right ) \right )}{{\sl Y}_{n}\left (\sin \left ( x \right ) \right )} \right ) }}}} \left ( {{\rm e}^{{\frac {{{\sl J}_{n}\left (\sin \left ( x \right ) \right )}{\it \_C1}}{\sin \left ( x \right ) \left ( {{\sl J}_{n }\left (\sin \left ( x \right ) \right )}{{\sl Y}_{n+1}\left (\sin \left ( x \right ) \right )}-{{\sl J}_{n+1}\left (\sin \left ( x \right ) \right )}{ {\sl Y}_{n}\left (\sin \left ( x \right ) \right )} \right ) }}}} \right ) ^ {-1}} \right \} \]