ODE
\[ y(x) \left (a \cot ^2(x)+b \cot (x) \csc (x)+c \csc ^2(x)\right )+k \cot (x) y'(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 45.1406 (sec), leaf count = 1
\[\text {$\$$Aborted}\]
Maple ✓
cpu = 2.546 (sec), leaf count = 441
\[\left [y \left (x \right ) = \textit {\_C1} \left (\sin ^{-\frac {k}{2}-\frac {1}{2}}\left (x \right )\right ) \hypergeom \left (\left [\frac {\sqrt {k^{2}-4 a -4 b -4 c -2 k +1}}{4}-\frac {\sqrt {k^{2}-4 a}}{2}-\frac {\sqrt {k^{2}-4 a +4 b -4 c -2 k +1}}{4}+\frac {1}{2}, \frac {\sqrt {k^{2}-4 a -4 b -4 c -2 k +1}}{4}+\frac {\sqrt {k^{2}-4 a}}{2}-\frac {\sqrt {k^{2}-4 a +4 b -4 c -2 k +1}}{4}+\frac {1}{2}\right ], \left [1-\frac {\sqrt {k^{2}-4 a +4 b -4 c -2 k +1}}{2}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) \sqrt {-2 \cos \left (x \right )+2}\, \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {k^{2}-4 a -4 b -4 c -2 k +1}}{4}} \left (2 \cos \left (x \right )+2\right )^{\frac {1}{2}-\frac {\sqrt {k^{2}-4 a +4 b -4 c -2 k +1}}{4}}+\textit {\_C2} \left (\sin ^{-\frac {k}{2}-\frac {1}{2}}\left (x \right )\right ) \hypergeom \left (\left [\frac {\sqrt {k^{2}-4 a -4 b -4 c -2 k +1}}{4}+\frac {\sqrt {k^{2}-4 a}}{2}+\frac {\sqrt {k^{2}-4 a +4 b -4 c -2 k +1}}{4}+\frac {1}{2}, \frac {\sqrt {k^{2}-4 a -4 b -4 c -2 k +1}}{4}-\frac {\sqrt {k^{2}-4 a}}{2}+\frac {\sqrt {k^{2}-4 a +4 b -4 c -2 k +1}}{4}+\frac {1}{2}\right ], \left [1+\frac {\sqrt {k^{2}-4 a +4 b -4 c -2 k +1}}{2}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) \sqrt {-2 \cos \left (x \right )+2}\, \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {k^{2}-4 a -4 b -4 c -2 k +1}}{4}} \left (2 \cos \left (x \right )+2\right )^{\frac {1}{2}+\frac {\sqrt {k^{2}-4 a +4 b -4 c -2 k +1}}{4}}\right ]\] Mathematica raw input
DSolve[(a*Cot[x]^2 + b*Cot[x]*Csc[x] + c*Csc[x]^2)*y[x] + k*Cot[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
$Aborted
Maple raw input
dsolve(diff(diff(y(x),x),x)+k*cot(x)*diff(y(x),x)+(a*cot(x)^2+b*cot(x)*csc(x)+c*csc(x)^2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*sin(x)^(-1/2*k-1/2)*hypergeom([1/4*(k^2-4*a-4*b-4*c-2*k+1)^(1/2)-1/2*(k^2-4*a)^(1/2)-1/4*(k^2-4*a+4*b-4*c-2*k+1)^(1/2)+1/2, 1/4*(k^2-4*a-4*b-4*c-2*k+1)^(1/2)+1/2*(k^2-4*a)^(1/2)-1/4*(k^2-4*a+4*b-4*c-2*k+1)^(1/2)+1/2],[1-1/2*(k^2-4*a+4*b-4*c-2*k+1)^(1/2)],1/2*cos(x)+1/2)*(-2*cos(x)+2)^(1/2)*(1/2*cos(x)-1/2)^(1/4*(k^2-4*a-4*b-4*c-2*k+1)^(1/2))*(2*cos(x)+2)^(1/2-1/4*(k^2-4*a+4*b-4*c-2*k+1)^(1/2))+_C2*sin(x)^(-1/2*k-1/2)*hypergeom([1/4*(k^2-4*a-4*b-4*c-2*k+1)^(1/2)+1/2*(k^2-4*a)^(1/2)+1/4*(k^2-4*a+4*b-4*c-2*k+1)^(1/2)+1/2, 1/4*(k^2-4*a-4*b-4*c-2*k+1)^(1/2)-1/2*(k^2-4*a)^(1/2)+1/4*(k^2-4*a+4*b-4*c-2*k+1)^(1/2)+1/2],[1+1/2*(k^2-4*a+4*b-4*c-2*k+1)^(1/2)],1/2*cos(x)+1/2)*(-2*cos(x)+2)^(1/2)*(1/2*cos(x)-1/2)^(1/4*(k^2-4*a-4*b-4*c-2*k+1)^(1/2))*(2*cos(x)+2)^(1/2+1/4*(k^2-4*a+4*b-4*c-2*k+1)^(1/2))]