##### 4.20.42 $$(y(x)+x)^2 y'(x)^2=y(x)^2$$

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

[[_homogeneous, class A], _rational, [_Abel, 2nd type, class A]]

Book solution method
No Missing Variables ODE, Solve for $$y'$$

Mathematica
cpu = 0.201307 (sec), leaf count = 64

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

Maple
cpu = 0.44 (sec), leaf count = 48

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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