4.3.41 \(2 y'(x)+2 \csc ^2(x)=y(x) \csc (x) \sec (x)-y(x)^2 \sec ^2(x)\)

ODE
\[ 2 y'(x)+2 \csc ^2(x)=y(x) \csc (x) \sec (x)-y(x)^2 \sec ^2(x) \] ODE Classification

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.476711 (sec), leaf count = 48

\[\left \{\left \{y(x)\to \frac {\cot (x) \left (c_1 \sqrt {\cos (x)}+2 \sqrt [4]{\sin ^2(x)}\right )}{c_1 \sqrt {\cos (x)}+\sqrt [4]{\sin ^2(x)}}\right \}\right \}\]

Maple
cpu = 0.09 (sec), leaf count = 107

\[ \left \{ \int ^{{\frac { \left (\csc \relax (x ) \sec \relax (x ) +2\,\cot \relax (x ) +2\,\tan \relax (x ) \right ) y \relax (x ) }{2\, \left (\csc \relax (x ) \right ) ^{2}}}}\!{ \left (\csc \relax (x ) \sec \relax (x ) +2\,\cot \relax (x ) +2\,\tan \relax (x ) \right ) ^{2} \left (2\, \left ({{\it \_a}}^{2}-{\it \_a}/2+1/2 \right ) \left (\sec \relax (x ) \right ) ^{2} \left (\csc \relax (x ) \right ) ^{2}-4\,\sec \relax (x ) \left (\cot \relax (x ) +\tan \relax (x ) \right ) \left ({\it \_a}-1 \right ) \csc \relax (x ) -4\, \left (\cot \relax (x ) +\tan \relax (x ) \right ) ^{2} \left ({\it \_a}-1 \right ) \right ) ^{-1}}{d{\it \_a}}+{\frac {\ln \left (\tan \relax (x ) \right ) }{2}}+\ln \left (\sin \relax (x ) \right ) -\ln \left (\cos \relax (x ) \right ) +{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[2*Csc[x]^2 + 2*y'[x] == Csc[x]*Sec[x]*y[x] - Sec[x]^2*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> (Cot[x]*(C[1]*Sqrt[Cos[x]] + 2*(Sin[x]^2)^(1/4)))/(C[1]*Sqrt[Cos[x]] +
 (Sin[x]^2)^(1/4))}}

Maple raw input

dsolve(2*diff(y(x),x)+2*csc(x)^2 = y(x)*csc(x)*sec(x)-y(x)^2*sec(x)^2, y(x),'implicit')

Maple raw output

Intat((csc(x)*sec(x)+2*cot(x)+2*tan(x))^2/(2*(_a^2-1/2*_a+1/2)*sec(x)^2*csc(x)^2
-4*sec(x)*(cot(x)+tan(x))*(_a-1)*csc(x)-4*(cot(x)+tan(x))^2*(_a-1)),_a = 1/2/csc
(x)^2*(csc(x)*sec(x)+2*cot(x)+2*tan(x))*y(x))+1/2*ln(tan(x))+ln(sin(x))-ln(cos(x
))+_C1 = 0