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 = 0.846388 (sec), leaf count = 48

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

Maple ✓
cpu = 0.113 (sec), leaf count = 176

\[\left [y \left (x \right ) = \frac {2 \RootOf \left (2 \left (\int _{}^{\textit {\_Z}}\frac {\left (\csc \left (x \right ) \sec \left (x \right )+2 \tan \left (x \right )+2 \cot \left (x \right )\right )^{2}}{2 \textit {\_a}^{2} \left (\csc ^{2}\left (x \right )\right ) \left (\sec ^{2}\left (x \right )\right )-\left (\csc ^{2}\left (x \right )\right ) \left (\sec ^{2}\left (x \right )\right ) \textit {\_a} +\left (\sec ^{2}\left (x \right )\right ) \left (\csc ^{2}\left (x \right )\right )-4 \csc \left (x \right ) \sec \left (x \right ) \tan \left (x \right ) \textit {\_a} -4 \csc \left (x \right ) \sec \left (x \right ) \cot \left (x \right ) \textit {\_a} +4 \csc \left (x \right ) \tan \left (x \right ) \sec \left (x \right )+4 \cot \left (x \right ) \csc \left (x \right ) \sec \left (x \right )-4 \left (\tan ^{2}\left (x \right )\right ) \textit {\_a} -8 \tan \left (x \right ) \cot \left (x \right ) \textit {\_a} -4 \left (\cot ^{2}\left (x \right )\right ) \textit {\_a} +4 \left (\tan ^{2}\left (x \right )\right )+8 \tan \left (x \right ) \cot \left (x \right )+4 \left (\cot ^{2}\left (x \right )\right )}d \textit {\_a} \right )+\ln \left (\tan \left (x \right )\right )+2 \ln \left (\sin \left (x \right )\right )-2 \ln \left (\cos \left (x \right )\right )+2 \textit {\_C1} \right ) \left (\csc ^{2}\left (x \right )\right )}{\csc \left (x \right ) \sec \left (x \right )+2 \tan \left (x \right )+2 \cot \left (x \right )}\right ]\] 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))

Maple raw output

[y(x) = 2*RootOf(2*Intat((csc(x)*sec(x)+2*tan(x)+2*cot(x))^2/(2*_a^2*csc(x)^2*se
c(x)^2-csc(x)^2*sec(x)^2*_a+sec(x)^2*csc(x)^2-4*csc(x)*sec(x)*tan(x)*_a-4*csc(x)
*sec(x)*cot(x)*_a+4*csc(x)*tan(x)*sec(x)+4*cot(x)*csc(x)*sec(x)-4*tan(x)^2*_a-8*
tan(x)*cot(x)*_a-4*cot(x)^2*_a+4*tan(x)^2+8*tan(x)*cot(x)+4*cot(x)^2),_a = _Z)+l
n(tan(x))+2*ln(sin(x))-2*ln(cos(x))+2*_C1)*csc(x)^2/(csc(x)*sec(x)+2*tan(x)+2*co
t(x))]