ODE
\[ a \left (x y'(x)-y(x)\right )+\log \left (y'(x)\right )=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.0403451 (sec), leaf count = 21
\[\left \{\left \{y(x)\to e^{-c_1} x-\frac {c_1}{a}\right \}\right \}\]
Maple ✓
cpu = 0.034 (sec), leaf count = 36
\[ \left \{ y \left ( x \right ) -{\frac {1}{a}\ln \left ( -{\frac {1}{ax}} \right ) }+{a}^{-1}=0,y \left ( x \right ) ={\it \_C1}\,x+{\frac {\ln \left ( {\it \_C1} \right ) }{a}} \right \} \] 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