4.24.4 a+by(x)+xy(x)+log(y(x))=0

ODE
a+by(x)+xy(x)+log(y(x))=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 = 0.149511 (sec), leaf count = 57

Solve[bW(xeaby(x))+(b+1)log(1bW(xeaby(x)))+b2y(x)b=c1,y(x)]

Maple
cpu = 0.012 (sec), leaf count = 88

{[x(_T)=_C1b_T+_T(b+1)1_T(b+1)1b_T,y(_T)=_T(b+1)1b_T(b+1)1ln(_T)+(a_T(b+1)1_C1_T)bb2_T(b+1)1]} 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^2*y[x])/b == 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),'implicit')

Maple raw output

[x(_T) = (_C1*b*_T+_T^(1/(b+1)))/(_T^(1/(b+1)))/b/_T, y(_T) = (-_T^(1/(b+1))-b*_
T^(1/(b+1))*ln(_T)+(-a*_T^(1/(b+1))-_C1*_T)*b)/b^2/(_T^(1/(b+1)))]