4.24.6 $a(x{y}^{\prime }(x)-y(x))+\log (y^{\prime }(x))=0$

ODE
$$a(x{y}^{\prime }(x)-y(x))+\log (y^{\prime }(x))=0$$ ODE Classification

[[_1st_order, _with_linear_symmetries], _Clairaut]

Book solution method
Clairaut’s equation and related types, $f(y-x{y}^{\prime },y^{\prime })=0$

Mathematica ✓
cpu = 0.0403451 (sec), leaf count = 21

$$\left\{\{y(x)\to e^{-c_1}x-\frac{c_1}{a}\}\right\}$$

Maple ✓
cpu = 0.034 (sec), leaf count = 36

$$\{y\left(x\right)-\frac{1}{a}\ln (-\frac{1}{ax})+a^{-1}=0,y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}x+\frac{\ln \left(\mathit{\_}\mathit{C}\mathit{1}\right)}{a}\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> x/E^C[1] - C[1]/a}}

Maple raw input

dsolve(ln(diff(y(x),x))+a*(x*diff(y(x),x)-y(x)) = 0, y(x),'implicit')

Maple raw output

y(x)-1/a*ln(-1/a/x)+1/a = 0, y(x) = _C1*x+ln(_C1)/a