4.5.19 \(x y'(x)=y(x) (-\log (y(x))+\log (x)+1)\)

ODE
\[ x y'(x)=y(x) (-\log (y(x))+\log (x)+1) \] ODE Classification

[[_homogeneous, `class A`], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.014644 (sec), leaf count = 17

\[\left \{\left \{y(x)\to x e^{\frac {e^{c_1}}{x}}\right \}\right \}\]

Maple
cpu = 0.02 (sec), leaf count = 17

\[ \left \{ x\ln \relax (x ) -x\ln \left (y \relax (x ) \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x*y'[x] == (1 + Log[x] - Log[y[x]])*y[x],y[x],x]

Mathematica raw output

{{y[x] -> E^(E^C[1]/x)*x}}

Maple raw input

dsolve(x*diff(y(x),x) = (1+ln(x)-ln(y(x)))*y(x), y(x),'implicit')

Maple raw output

x*ln(x)-x*ln(y(x))-_C1 = 0