4.18.5 $$x y'(x)^2+x y'(x)-y(x)=0$$

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

[[_homogeneous, class A], _rational, _dAlembert]

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

Mathematica
cpu = 0.450918 (sec), leaf count = 95

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

Maple
cpu = 0.182 (sec), leaf count = 69

$\left [y \left (x \right ) = \left (\frac {1}{4 \LambertW \left (-\frac {1}{2 \sqrt {\frac {\textit {\_C1}}{x}}}\right )^{2}}+\frac {1}{2 \LambertW \left (-\frac {1}{2 \sqrt {\frac {\textit {\_C1}}{x}}}\right )}\right ) x, y \left (x \right ) = \left (\frac {1}{4 \LambertW \left (\frac {1}{2 \sqrt {\frac {\textit {\_C1}}{x}}}\right )^{2}}+\frac {1}{2 \LambertW \left (\frac {1}{2 \sqrt {\frac {\textit {\_C1}}{x}}}\right )}\right ) x\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = (1/4/LambertW(-1/2/(1/x*_C1)^(1/2))^2+1/2/LambertW(-1/2/(1/x*_C1)^(1/2))
)*x, y(x) = (1/4/LambertW(1/2/(1/x*_C1)^(1/2))^2+1/2/LambertW(1/2/(1/x*_C1)^(1/2
)))*x]