ODE
\[ x \left (x^2+y(x)^2\right ) y'(x)=y(x) \left (x^4+x^2+y(x)^2\right ) \] ODE Classification
[[_homogeneous, `class D`], _rational]
Book solution method
Homogeneous equation, \(xy'(x)=x f(x) g(u)+y(x)\)
Mathematica ✓
cpu = 0.0199941 (sec), leaf count = 44
\[\left \{\left \{y(x)\to -x \sqrt {W\left (e^{2 c_1+x^2}\right )}\right \},\left \{y(x)\to x \sqrt {W\left (e^{2 c_1+x^2}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.011 (sec), leaf count = 28
\[ \left \{ \ln \left ( {\frac {y \left ( x \right ) }{x}} \right ) +{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{2\,{x}^{2}}}-{\frac {{x}^{2}}{2}}-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*(x^2 + y[x]^2)*y'[x] == y[x]*(x^2 + x^4 + y[x]^2),y[x],x]
Mathematica raw output
{{y[x] -> -(x*Sqrt[ProductLog[E^(x^2 + 2*C[1])]])}, {y[x] -> x*Sqrt[ProductLog[E^(x^2 + 2*C[1])]]}}
Maple raw input
dsolve(x*(x^2+y(x)^2)*diff(y(x),x) = (x^2+x^4+y(x)^2)*y(x), y(x),'implicit')
Maple raw output
ln(y(x)/x)+1/2/x^2*y(x)^2-1/2*x^2-_C1 = 0