ODE
\[ y''(x)-\csc (2 x) y'(x)+y(x) \left (\sin ^2(x)+2\right ) \csc ^2(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.810862 (sec), leaf count = 146
\[\left \{\left \{y(x)\to \frac {\left (-\sin ^2(x)\right )^{\frac {1}{2}-\frac {i \sqrt {23}}{8}} \cos ^2(x)^{3/8} \left (c_1 \, _2F_1\left (-\frac {1}{8}-\frac {i \sqrt {23}}{8},\frac {7}{8}-\frac {i \sqrt {23}}{8};\frac {3}{4};\cos ^2(x)\right )+\sqrt [4]{-1} c_2 \sqrt [4]{\cos ^2(x)} \, _2F_1\left (\frac {1}{8}-\frac {i \sqrt {23}}{8},\frac {9}{8}-\frac {i \sqrt {23}}{8};\frac {5}{4};\cos ^2(x)\right )\right )}{\sqrt [8]{\sin ^2(x)} \cos ^{\frac {3}{4}}(x)}\right \}\right \}\]
Maple ✓
cpu = 1.735 (sec), leaf count = 101
\[\left [y \left (x \right ) = \textit {\_C1} \left (\sin ^{\frac {3}{4}-\frac {i \sqrt {23}}{4}}\left (x \right )\right ) \left (\sqrt {\cos }\left (x \right )\right ) \hypergeom \left (\left [\frac {9}{8}-\frac {i \sqrt {23}}{8}, \frac {1}{8}-\frac {i \sqrt {23}}{8}\right ], \left [1-\frac {i \sqrt {23}}{4}\right ], \sin ^{2}\left (x \right )\right )+\textit {\_C2} \left (\sin ^{\frac {3}{4}+\frac {i \sqrt {23}}{4}}\left (x \right )\right ) \left (\sqrt {\cos }\left (x \right )\right ) \hypergeom \left (\left [\frac {9}{8}+\frac {i \sqrt {23}}{8}, \frac {1}{8}+\frac {i \sqrt {23}}{8}\right ], \left [1+\frac {i \sqrt {23}}{4}\right ], \sin ^{2}\left (x \right )\right )\right ]\] Mathematica raw input
DSolve[Csc[x]^2*(2 + Sin[x]^2)*y[x] - Csc[2*x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((Cos[x]^2)^(3/8)*(C[1]*Hypergeometric2F1[-1/8 - (I/8)*Sqrt[23], 7/8 - (I/8)*Sqrt[23], 3/4, Cos[x]^2] + (-1)^(1/4)*C[2]*(Cos[x]^2)^(1/4)*Hypergeometric2F1[1/8 - (I/8)*Sqrt[23], 9/8 - (I/8)*Sqrt[23], 5/4, Cos[x]^2])*(-Sin[x]^2)^(1/2 - (I/8)*Sqrt[23]))/(Cos[x]^(3/4)*(Sin[x]^2)^(1/8))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-diff(y(x),x)*csc(2*x)+csc(x)^2*(2+sin(x)^2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*sin(x)^(3/4-1/4*I*23^(1/2))*cos(x)^(1/2)*hypergeom([9/8-1/8*I*23^(1/2), 1/8-1/8*I*23^(1/2)],[1-1/4*I*23^(1/2)],sin(x)^2)+_C2*sin(x)^(3/4+1/4*I*23^(1/2))*cos(x)^(1/2)*hypergeom([9/8+1/8*I*23^(1/2), 1/8+1/8*I*23^(1/2)],[1+1/4*I*23^(1/2)],sin(x)^2)]