4.3.29 $$y'(x)+\csc (2 x) \sin (2 y(x))=0$$

ODE
$y'(x)+\csc (2 x) \sin (2 y(x))=0$ ODE Classiﬁcation

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.380207 (sec), leaf count = 15

$\left \{\left \{y(x)\to \cot ^{-1}\left (e^{-2 c_1} \tan (x)\right )\right \}\right \}$

Maple
cpu = 0.172 (sec), leaf count = 105

$\left [y \left (x \right ) = \frac {\arctan \left (-\frac {2 \textit {\_C1} \left (\sin \left (4 x \right )+2 \sin \left (2 x \right )\right )}{\textit {\_C1}^{2} \cos \left (4 x \right )-\textit {\_C1}^{2}-\cos \left (4 x \right )-4 \cos \left (2 x \right )-3}, \frac {\textit {\_C1}^{2} \cos \left (4 x \right )-\textit {\_C1}^{2}+\cos \left (4 x \right )+4 \cos \left (2 x \right )+3}{\textit {\_C1}^{2} \cos \left (4 x \right )-\textit {\_C1}^{2}-\cos \left (4 x \right )-4 \cos \left (2 x \right )-3}\right )}{2}\right ]$ Mathematica raw input

DSolve[Csc[2*x]*Sin[2*y[x]] + y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> ArcCot[Tan[x]/E^(2*C[1])]}}

Maple raw input

dsolve(diff(y(x),x)+csc(2*x)*sin(2*y(x)) = 0, y(x))

Maple raw output

[y(x) = 1/2*arctan(-2*_C1*(sin(4*x)+2*sin(2*x))/(_C1^2*cos(4*x)-_C1^2-cos(4*x)-4
*cos(2*x)-3),(_C1^2*cos(4*x)-_C1^2+cos(4*x)+4*cos(2*x)+3)/(_C1^2*cos(4*x)-_C1^2-
cos(4*x)-4*cos(2*x)-3))]