4.12.32 \(x^2 (x-2 y(x)) y'(x)=2 x^3-4 x y(x)^2+y(x)^3\)

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

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type``class C`], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0656077 (sec), leaf count = 101

\[\left \{\left \{y(x)\to \frac {2 x^3-\sqrt {e^{2 c_1} x^2 \left (e^{2 c_1}-3 x^2\right )}}{e^{2 c_1}+x^2}\right \},\left \{y(x)\to \frac {\sqrt {e^{2 c_1} x^2 \left (e^{2 c_1}-3 x^2\right )}+2 x^3}{e^{2 c_1}+x^2}\right \}\right \}\]

Maple
cpu = 0.03 (sec), leaf count = 48

\[ \left \{ {\frac {1}{2}\ln \left ({\frac {y \relax (x ) -x}{x}} \right ) }-\ln \left ({\frac {y \relax (x ) -2\,x}{x}} \right ) +{\frac {1}{2}\ln \left ({\frac {x+y \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (2*x^3 - Sqrt[E^(2*C[1])*x^2*(E^(2*C[1]) - 3*x^2)])/(E^(2*C[1]) + x^2)
}, {y[x] -> (2*x^3 + Sqrt[E^(2*C[1])*x^2*(E^(2*C[1]) - 3*x^2)])/(E^(2*C[1]) + x^
2)}}

Maple raw input

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

Maple raw output

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