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