4.12.32 $x^2(x-2y(x))y^{\prime }(x)=2x^3-4xy(x{)}^2+y(x{)}^3$

ODE
$$x^2(x-2y(x))y^{\prime }(x)=2x^3-4xy(x{)}^2+y(x{)}^3$$ 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

$$\{\{y(x)\to \frac{2x^3-\sqrt{e^{2c_1}{x}^2(e^{2c_1}-3x^2)}}{e^{2c_1}+x^2}\},\{y(x)\to \frac{\sqrt{e^{2c_1}{x}^2(e^{2c_1}-3x^2)}+2x^3}{e^{2c_1}+x^2}\}\}$$

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

$$\{\frac{1}{2}\ln \left(\frac{y\left(x\right)-x}{x}\right)-\ln \left(\frac{y\left(x\right)-2x}{x}\right)+\frac{1}{2}\ln \left(\frac{x+y\left(x\right)}{x}\right)-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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