ODE
\[ y''(x)+\cot (x) y'(x)-y(x) \csc ^2(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0584413 (sec), leaf count = 51
\[\left \{\left \{y(x)\to c_1 \cosh \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )-i c_2 \sinh \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.033 (sec), leaf count = 25
\[ \left \{ y \left ( x \right ) ={\frac {\sin \left ( x \right ) {\it \_C1}}{\cos \left ( x \right ) -1}}+{\frac { \left ( \cos \left ( x \right ) -1 \right ) {\it \_C2}}{\sin \left ( x \right ) }} \right \} \] Mathematica raw input
DSolve[-(Csc[x]^2*y[x]) + Cot[x]*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cosh[Log[Cos[x/2]] - Log[Sin[x/2]]] - I*C[2]*Sinh[Log[Cos[x/2]] - Log[Sin[x/2]]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+cot(x)*diff(y(x),x)-y(x)*csc(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x) = sin(x)/(cos(x)-1)*_C1+(cos(x)-1)/sin(x)*_C2