4.13.46 \(x \left (2 x^2+y(x)^2\right ) y'(x)=y(x) \left (2 x^2+3 y(x)^2\right )\)

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

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0379948 (sec), leaf count = 56

\[\left \{\left \{y(x)\to -\frac {\sqrt {2} x}{\sqrt {W\left (\frac {2 e^{-2 c_1}}{x^4}\right )}}\right \},\left \{y(x)\to \frac {\sqrt {2} x}{\sqrt {W\left (\frac {2 e^{-2 c_1}}{x^4}\right )}}\right \}\right \}\]

Maple
cpu = 0.018 (sec), leaf count = 29

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

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

Mathematica raw output

{{y[x] -> -((Sqrt[2]*x)/Sqrt[ProductLog[2/(E^(2*C[1])*x^4)]])}, {y[x] -> (Sqrt[2
]*x)/Sqrt[ProductLog[2/(E^(2*C[1])*x^4)]]}}

Maple raw input

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

Maple raw output

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