##### 4.10.40 $$4 (-y(x)-x+1) y'(x)-x+2=0$$

ODE
$4 (-y(x)-x+1) y'(x)-x+2=0$ ODE Classiﬁcation

[[_homogeneous, class C], _rational, [_Abel, 2nd type, class C], _dAlembert]

Book solution method
Equation linear in the variables, $$y'(x)=f\left ( \frac {X_1}{X_2} \right )$$

Mathematica
cpu = 6.12883 (sec), leaf count = 109

$\text {Solve}\left [\frac {2^{2/3} \left (x \log \left (\frac {x-2}{y(x)+x-1}\right )-x \log \left (\frac {2 y(x)+x}{y(x)+x-1}\right )+2 y(x) \left (\log \left (\frac {x-2}{y(x)+x-1}\right )-\log \left (\frac {2 y(x)+x}{y(x)+x-1}\right )+1\right )+2 x-2\right )}{9 (2 y(x)+x)}=\frac {1}{9} 2^{2/3} \log (x-2)+c_1,y(x)\right ]$

Maple
cpu = 0.128 (sec), leaf count = 29

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

DSolve[2 - x + 4*(1 - x - y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[(2^(2/3)*(-2 + 2*x + x*Log[(-2 + x)/(-1 + x + y[x])] - x*Log[(x + 2*y[x])/
(-1 + x + y[x])] + 2*(1 + Log[(-2 + x)/(-1 + x + y[x])] - Log[(x + 2*y[x])/(-1 +
 x + y[x])])*y[x]))/(9*(x + 2*y[x])) == C[1] + (2^(2/3)*Log[-2 + x])/9, y[x]]

Maple raw input

dsolve(4*(1-x-y(x))*diff(y(x),x)+2-x = 0, y(x))

Maple raw output

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