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

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

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type``class C`]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0130467 (sec), leaf count = 74

\[\left \{\left \{y(x)\to \frac {x}{\frac {\sqrt {x \left (c_1-2 \log (x)+4\right )}}{\sqrt {\frac {1}{x^3}}}-2 x^2}\right \},\left \{y(x)\to -\frac {x}{\frac {\sqrt {x \left (c_1-2 \log (x)+4\right )}}{\sqrt {\frac {1}{x^3}}}+2 x^2}\right \}\right \}\]

Maple
cpu = 0.018 (sec), leaf count = 25

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

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

Mathematica raw output

{{y[x] -> x/(-2*x^2 + Sqrt[x*(4 + C[1] - 2*Log[x])]/Sqrt[x^(-3)])}, {y[x] -> -(x
/(2*x^2 + Sqrt[x*(4 + C[1] - 2*Log[x])]/Sqrt[x^(-3)]))}}

Maple raw input

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

Maple raw output

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