##### 4.16.49 $$y'(x)^2+2 (1-x) y'(x)-2 (x-y(x))=0$$

ODE
$y'(x)^2+2 (1-x) y'(x)-2 (x-y(x))=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries], _dAlembert]

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

Mathematica
cpu = 0.959945 (sec), leaf count = 99

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

Maple
cpu = 0.046 (sec), leaf count = 45

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

DSolve[-2*(x - y[x]) + 2*(1 - x)*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(diff(y(x),x)^2+2*(1-x)*diff(y(x),x)-2*x+2*y(x) = 0, y(x))

Maple raw output

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