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 = 0.34607 (sec), leaf count = 20
\[\left \{\left \{y(x)\to (c_1-i \pi ) x+e^{c_1}\right \}\right \}\]
Maple ✓
cpu = 0.464 (sec), leaf count = 23
\[[y \left (x \right ) = x \ln \left (x \right )-x, y \left (x \right ) = -{\mathrm e}^{\textit {\_C1}}+\textit {\_C1} x]\] 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))
Maple raw output
[y(x) = x*ln(x)-x, y(x) = -exp(_C1)+_C1*x]