4.3.19 \(y'(x)=\sin (x) (\csc (y(x))-\cot (y(x)))\)

ODE
\[ y'(x)=\sin (x) (\csc (y(x))-\cot (y(x))) \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.112868 (sec), leaf count = 21

\[\left \{\left \{y(x)\to 2 \sin ^{-1}\left (e^{\frac {1}{4} \left (c_1-2 \cos (x)\right )}\right )\right \}\right \}\]

Maple
cpu = 0.042 (sec), leaf count = 15

\[ \left \{ {\it \_C1}+\cos \relax (x ) +\ln \left (1-\cos \left (y \relax (x ) \right ) \right ) =0 \right \} \] Mathematica raw input

DSolve[y'[x] == (-Cot[y[x]] + Csc[y[x]])*Sin[x],y[x],x]

Mathematica raw output

{{y[x] -> 2*ArcSin[E^((C[1] - 2*Cos[x])/4)]}}

Maple raw input

dsolve(diff(y(x),x) = sin(x)*(csc(y(x))-cot(y(x))), y(x),'implicit')

Maple raw output

_C1+cos(x)+ln(1-cos(y(x))) = 0