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 = 0.394823 (sec), leaf count = 44
\[\left \{\left \{y(x)\to -x \sqrt {W\left (e^{x^2+2 c_1}\right )}\right \},\left \{y(x)\to x \sqrt {W\left (e^{x^2+2 c_1}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.104 (sec), leaf count = 26
\[\left [y \left (x \right ) = {\mathrm e}^{-\frac {\LambertW \left ({\mathrm e}^{x^{2}} {\mathrm e}^{2 \textit {\_C1}}\right )}{2}+\frac {x^{2}}{2}+\textit {\_C1}} x\right ]\] 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))
Maple raw output
[y(x) = exp(-1/2*LambertW(exp(x^2)*exp(_C1)^2)+1/2*x^2+_C1)*x]