4.14.10 \(x \left (-3 x^3 y(x)+x^3+4 y(x)^2\right ) y'(x)=6 y(x)^3\)

ODE
\[ x \left (-3 x^3 y(x)+x^3+4 y(x)^2\right ) y'(x)=6 y(x)^3 \] ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.397961 (sec), leaf count = 27

\[\text {Solve}\left [\frac {y(x)^2}{x^3}+\frac {1}{2} (\log (y(x))-3 y(x))=c_1,y(x)\right ]\]

Maple
cpu = 0.112 (sec), leaf count = 23

\[ \left \{ {\it \_C1}-{\frac {y \relax (x ) }{2}}+{\frac {\ln \left (y \relax (x ) \right ) }{6}}+{\frac { \left (y \relax (x ) \right ) ^{2}}{3\,{x}^{3}}}=0 \right \} \] Mathematica raw input

DSolve[x*(x^3 - 3*x^3*y[x] + 4*y[x]^2)*y'[x] == 6*y[x]^3,y[x],x]

Mathematica raw output

Solve[(Log[y[x]] - 3*y[x])/2 + y[x]^2/x^3 == C[1], y[x]]

Maple raw input

dsolve(x*(x^3-3*x^3*y(x)+4*y(x)^2)*diff(y(x),x) = 6*y(x)^3, y(x),'implicit')

Maple raw output

_C1-1/2*y(x)+1/6*ln(y(x))+1/3/x^3*y(x)^2 = 0