4.2.23 \(y'(x)+4 \csc (x)=y(x)^2 \sin (x)+y(x) (3-\cot (x))\)

ODE
\[ y'(x)+4 \csc (x)=y(x)^2 \sin (x)+y(x) (3-\cot (x)) \] ODE Classification

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.0565983 (sec), leaf count = 24

\[\left \{\left \{y(x)\to \left (\frac {1}{c_1 e^{5 x}+\frac {1}{5}}-4\right ) \csc (x)\right \}\right \}\]

Maple
cpu = 0.273 (sec), leaf count = 35

\[ \left \{ y \relax (x ) ={\frac {{\it \_C1}\,{{\rm e}^{-x}}-4\,{{\rm e}^{4\,x}}}{\sin \relax (x ) \left ({\it \_C1}\,{{\rm e}^{-x}}+{{\rm e}^{4\,x}} \right ) }} \right \} \] Mathematica raw input

DSolve[4*Csc[x] + y'[x] == (3 - Cot[x])*y[x] + Sin[x]*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> (-4 + (1/5 + E^(5*x)*C[1])^(-1))*Csc[x]}}

Maple raw input

dsolve(diff(y(x),x)+4*csc(x) = (3-cot(x))*y(x)+y(x)^2*sin(x), y(x),'implicit')

Maple raw output

y(x) = (_C1*exp(-x)-4*exp(4*x))/sin(x)/(_C1*exp(-x)+exp(4*x))