4.15.4 \(x \left (-2 x y(x)^3-x y(x)^2+2\right ) y'(x)+2 y(x)+1=0\)

ODE
\[ x \left (-2 x y(x)^3-x y(x)^2+2\right ) y'(x)+2 y(x)+1=0 \] ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.280131 (sec), leaf count = 42

\[\text {Solve}\left [\frac {1}{64} \left (-4 y(x)^2+4 y(x)-\frac {16}{2 x y(x)+x}-2 \log (8 y(x)+4)+3\right )=c_1,y(x)\right ]\]

Maple
cpu = 0.139 (sec), leaf count = 33

\[ \left \{ \left (2\,xy \relax (x ) +x \right ) ^{-1}+{\frac { \left (y \relax (x ) \right ) ^{2}}{4}}-{\frac {y \relax (x ) }{4}}+{\frac {\ln \left (1+2\,y \relax (x ) \right ) }{8}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[1 + 2*y[x] + x*(2 - x*y[x]^2 - 2*x*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[(3 - 2*Log[4 + 8*y[x]] + 4*y[x] - 4*y[x]^2 - 16/(x + 2*x*y[x]))/64 == C[1]
, y[x]]

Maple raw input

dsolve(x*(2-x*y(x)^2-2*x*y(x)^3)*diff(y(x),x)+1+2*y(x) = 0, y(x),'implicit')

Maple raw output

1/(2*x*y(x)+x)+1/4*y(x)^2-1/4*y(x)+1/8*ln(1+2*y(x))+_C1 = 0