##### 4.24.4 $$a+b y(x)+x y'(x)+\log \left (y'(x)\right )=0$$

ODE
$a+b y(x)+x y'(x)+\log \left (y'(x)\right )=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries], _dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 0.376569 (sec), leaf count = 52

$\text {Solve}\left [W\left (x e^{-a-b y(x)}\right )+\frac {(b+1) \log \left (1-b W\left (x e^{-a-b y(x)}\right )\right )}{b}+b y(x)=c_1,y(x)\right ]$

Maple
cpu = 2.127 (sec), leaf count = 66

$\left [-\left ({\mathrm e}^{-b y \left (x \right )-\LambertW \left (x \,{\mathrm e}^{-b y \left (x \right )-a}\right )-a}\right )^{-\frac {1}{b +1}} \textit {\_C1} +x -\frac {{\mathrm e}^{b y \left (x \right )+\LambertW \left (x \,{\mathrm e}^{-b y \left (x \right )-a}\right )+a}}{b} = 0\right ]$ Mathematica raw input

DSolve[a + Log[y'[x]] + b*y[x] + x*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[((1 + b)*Log[1 - b*ProductLog[E^(-a - b*y[x])*x]])/b + ProductLog[E^(-a -
b*y[x])*x] + b*y[x] == C[1], y[x]]

Maple raw input

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

Maple raw output

[-exp(-b*y(x)-LambertW(x*exp(-b*y(x)-a))-a)^(-1/(b+1))*_C1+x-1/b*exp(b*y(x)+Lamb
ertW(x*exp(-b*y(x)-a))+a) = 0]