4.38.31 $x^4{y}^{″}(x)=x(x^2+2y(x))y^{\prime }(x)-4y(x{)}^2$

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

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0702465 (sec), leaf count = 83

$$\left\{\{y(x)\to \frac{x^2((1-i\sqrt{-c_1-1})x^{2i\sqrt{-c_1-1}}+(1+i\sqrt{-c_1-1})c_2)}{c_2+x^{2i\sqrt{-c_1-1}}}\}\right\}$$

Maple ✓
cpu = 0.058 (sec), leaf count = 26

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{2}-\mathit{\_}\mathit{C}\mathit{1}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\frac{(x^2-y\left(x\right))\mathit{\_}\mathit{C}\mathit{1}}{x^2}\right)=0\}$$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

ln(x)-_C2-_C1*arctanh((x^2-y(x))/x^2*_C1) = 0