4.14.48 \(x \left (x^3+3 x^2 y(x)+y(x)^3\right ) y'(x)=y(x)^2 \left (3 x^2+y(x)^2\right )\)

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

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.065418 (sec), leaf count = 34

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

Maple
cpu = 0.014 (sec), leaf count = 36

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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