4.27.12 \(y''(x)+\cot (x) y'(x)-y(x) \csc ^2(x)=0\)

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