4.13.43 \(x \left (x^2+y(x)^2\right ) y'(x)=y(x) \left (x^4+x^2+y(x)^2\right )\)

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

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