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 = 67.4996 (sec), leaf count = 1
\[\text {$\$$Aborted}\]
Maple ✓
cpu = 0.632 (sec), leaf count = 398
\[ \left \{ y \left ( x \right ) = \left ( {\frac {\cos \left ( x \right ) }{2}}-{\frac {1}{2}} \right ) ^{{\frac {1}{4}\sqrt {{k}^{2}-4\,a-4\,b-4\,c-2\,k+1}}}\sqrt {2-2\,\cos \left ( x \right ) } \left ( \sin \left ( x \right ) \right ) ^{-{\frac {k}{2}}-{\frac {1}{2}}} \left ( {\mbox {$_2$F$_1$}({\frac {1}{4}\sqrt {{k}^{2}-4\,a-4\,b-4\,c-2\,k+1}}+{\frac {1}{2}\sqrt {{k}^{2}-4\,a}}+{\frac {1}{4}\sqrt {{k}^{2}-4\,a+4\,b-4\,c-2\,k+1}}+{\frac {1}{2}},{\frac {1}{4}\sqrt {{k}^{2}-4\,a-4\,b-4\,c-2\,k+1}}-{\frac {1}{2}\sqrt {{k}^{2}-4\,a}}+{\frac {1}{4}\sqrt {{k}^{2}-4\,a+4\,b-4\,c-2\,k+1}}+{\frac {1}{2}};\,1+{\frac {1}{2}\sqrt {{k}^{2}-4\,a+4\,b-4\,c-2\,k+1}};\,{\frac {\cos \left ( x \right ) }{2}}+{\frac {1}{2}})} \left ( 2\,\cos \left ( x \right ) +2 \right ) ^{{\frac {1}{2}}+{\frac {1}{4}\sqrt {{k}^{2}-4\,a+4\,b-4\,c-2\,k+1}}}{\it \_C2}+ \left ( 2\,\cos \left ( x \right ) +2 \right ) ^{{\frac {1}{2}}-{\frac {1}{4}\sqrt {{k}^{2}-4\,a+4\,b-4\,c-2\,k+1}}}{\mbox {$_2$F$_1$}({\frac {1}{4}\sqrt {{k}^{2}-4\,a-4\,b-4\,c-2\,k+1}}+{\frac {1}{2}\sqrt {{k}^{2}-4\,a}}-{\frac {1}{4}\sqrt {{k}^{2}-4\,a+4\,b-4\,c-2\,k+1}}+{\frac {1}{2}},{\frac {1}{4}\sqrt {{k}^{2}-4\,a-4\,b-4\,c-2\,k+1}}-{\frac {1}{2}\sqrt {{k}^{2}-4\,a}}-{\frac {1}{4}\sqrt {{k}^{2}-4\,a+4\,b-4\,c-2\,k+1}}+{\frac {1}{2}};\,1-{\frac {1}{2}\sqrt {{k}^{2}-4\,a+4\,b-4\,c-2\,k+1}};\,{\frac {\cos \left ( x \right ) }{2}}+{\frac {1}{2}})}{\it \_C1} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = (1/2*cos(x)-1/2)^(1/4*(k^2-4*a-4*b-4*c-2*k+1)^(1/2))*(2-2*cos(x))^(1/2)*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+1/4*(k^2-4*a+4*b-4*c-2*k+1)^(1/2))*_C2+(2*cos(x)+2)^(1/2-1/4*(k^2-4*a+4*b-4*c-2*k+1)^(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)*_C1)