4.141   2y′(x)+ 2csc2(x) = y(x)csc(x)sec(x)− y(x)2sec2(x)

ODE

2y′(x)+ 2csc2(x) = y(x)csc(x)sec(x)− y(x)2sec2(x)

ODE Classification

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

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

((             (                    ) ) )
||||                ∘ ------   4∘ --2--- || ||
{{        cot(x)--c1--cos(x-)+-2--sin-(x)-} }
|||| y(x) →       ∘ ------  4∘---2--     || ||
((            c1  cos(x) +  sin (x)     ) )

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

{∫ (csc(x)sec(x)+2cot(x)+2tan(x))y(x)                                                                                                                                                                                 }
   --------2(csc(x))2--------                               2 ( (  2           )       2       2                                                      2       )−1     ln(tan(x))
                          (csc(x)sec(x)+ 2cot(x)+ 2tan(x))  2 a  − a∕2 + 1∕2 (sec(x)) (csc(x)) − 4sec(x)(cot(x)+ tan(x))( a − 1)csc(x)− 4 (cot(

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