4.17.31 $$2 x y(x)^3 y'(x)+y'(x)^2+y(x)^4=0$$

ODE
$2 x y(x)^3 y'(x)+y'(x)^2+y(x)^4=0$ ODE Classiﬁcation

[[_homogeneous, class G]]

Book solution method
No Missing Variables ODE, Solve for $$x$$

Mathematica
cpu = 0.938854 (sec), leaf count = 136

$\left \{\text {Solve}\left [\frac {\sqrt {x^2 y(x)^2-1} y(x)^2 \tanh ^{-1}\left (\frac {x y(x)}{\sqrt {x^2 y(x)^2-1}}\right )}{\sqrt {y(x)^4 \left (x^2 y(x)^2-1\right )}}+\log (y(x))+c_1=0,y(x)\right ],\text {Solve}\left [\log (y(x))+c_1=\frac {y(x)^2 \sqrt {x^2 y(x)^2-1} \tanh ^{-1}\left (\frac {x y(x)}{\sqrt {x^2 y(x)^2-1}}\right )}{\sqrt {y(x)^4 \left (x^2 y(x)^2-1\right )}},y(x)\right ]\right \}$

Maple
cpu = 2.702 (sec), leaf count = 47

$\left [y \left (x \right ) = \frac {1}{x}, y \left (x \right ) = -\frac {1}{x}, y \left (x \right ) = \frac {1}{\sqrt {-\textit {\_C1}^{2}+2 x \textit {\_C1}}}, y \left (x \right ) = -\frac {1}{\sqrt {-\textit {\_C1}^{2}+2 x \textit {\_C1}}}\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[C[1] + Log[y[x]] + (ArcTanh[(x*y[x])/Sqrt[-1 + x^2*y[x]^2]]*y[x]^2*Sqrt[-
1 + x^2*y[x]^2])/Sqrt[y[x]^4*(-1 + x^2*y[x]^2)] == 0, y[x]], Solve[C[1] + Log[y[
x]] == (ArcTanh[(x*y[x])/Sqrt[-1 + x^2*y[x]^2]]*y[x]^2*Sqrt[-1 + x^2*y[x]^2])/Sq
rt[y[x]^4*(-1 + x^2*y[x]^2)], y[x]]}

Maple raw input

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

Maple raw output

[y(x) = 1/x, y(x) = -1/x, y(x) = 1/(-_C1^2+2*_C1*x)^(1/2), y(x) = -1/(-_C1^2+2*_
C1*x)^(1/2)]