ODE
\[ \log \left (x y'(x)-y(x)\right )=y'(x) \] ODE Classification
[[_1st_order, _with_linear_symmetries], _Clairaut]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 0.113049 (sec), leaf count = 20
\[\left \{\left \{y(x)\to e^{c_1}+\left (c_1-i \pi \right ) x\right \}\right \}\]
Maple ✓
cpu = 0.076 (sec), leaf count = 23
\[ \left \{ y \left ( x \right ) -x\ln \left ( x \right ) +x=0,y \left ( x \right ) =-{{\rm e}^{{\it \_C1}}}+{\it \_C1}\,x \right \} \] Mathematica raw input
DSolve[Log[-y[x] + x*y'[x]] == y'[x],y[x],x]
Mathematica raw output
{{y[x] -> E^C[1] + x*((-I)*Pi + C[1])}}
Maple raw input
dsolve(ln(x*diff(y(x),x)-y(x)) = diff(y(x),x), y(x),'implicit')
Maple raw output
y(x)-x*ln(x)+x = 0, y(x) = -exp(_C1)+_C1*x