4.19.39 $x^4{y}^{\prime }(x{)}^2+xy(x{)}^2{y}^{\prime }(x)-y(x{)}^3=0$

ODE
$$x^4{y}^{\prime }(x{)}^2+xy(x{)}^2{y}^{\prime }(x)-y(x{)}^3=0$$ ODE Classification

[[_homogeneous, `class G`]]

Book solution method
Change of variable

Mathematica ✓
cpu = 0.20814 (sec), leaf count = 50

$$\{\{y(x)\to -\frac{4e^{2c_1}x}{2e^{c_1}-x}\},\{y(x)\to \frac{e^{2c_1}x}{2e^{c_1}+4x}\}\}$$

Maple ✓
cpu = 0.177 (sec), leaf count = 84

$$\{2{\left(y\left(x\right)\right)}^{-1}+x^{-2}+\frac{1}{x^{2}y\left(x\right)}\sqrt{{\left(y\left(x\right)\right)}^2+4x^{2}y\left(x\right)}-\mathit{\_}\mathit{C}\mathit{1}=0,2y\left(x\right)+\frac{{\left(y\left(x\right)\right)}^2}{x^2}+\frac{y\left(x\right)}{x^2}\sqrt{{\left(y\left(x\right)\right)}^2+4x^{2}y\left(x\right)}-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=-4x^2\}$$ Mathematica raw input

DSolve[-y[x]^3 + x*y[x]^2*y'[x] + x^4*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-4*E^(2*C[1])*x)/(2*E^C[1] - x)}, {y[x] -> (E^(2*C[1])*x)/(2*E^C[1] +
 4*x)}}

Maple raw input

dsolve(x^4*diff(y(x),x)^2+x*y(x)^2*diff(y(x),x)-y(x)^3 = 0, y(x),'implicit')

Maple raw output

y(x) = -4*x^2, 2*y(x)+y(x)^2/x^2+y(x)/x^2*(y(x)^2+4*x^2*y(x))^(1/2)-_C1 = 0, 2/y
(x)+1/x^2+1/y(x)/x^2*(y(x)^2+4*x^2*y(x))^(1/2)-_C1 = 0