##### 4.10.9 $$y'(x) (a+b x+y(x))+a-b x-y(x)=0$$

ODE
$y'(x) (a+b x+y(x))+a-b x-y(x)=0$ ODE Classiﬁcation

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

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

Mathematica
cpu = 0.204905 (sec), leaf count = 49

$\left \{\left \{y(x)\to \frac {2 a W\left (-e^{\frac {(b+1)^2 x}{2 a}-1+c_1}\right )+a (-b)+a-b (b+1) x}{b+1}\right \}\right \}$

Maple
cpu = 0.233 (sec), leaf count = 91

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

DSolve[a - b*x - y[x] + (a + b*x + y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve((a+b*x+y(x))*diff(y(x),x)+a-b*x-y(x) = 0, y(x))

Maple raw output

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