ODE
\[ x y'(x) (a+b y(x))=c y(x) \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.180391 (sec), leaf count = 31
\[\left \{\left \{y(x)\to \frac {a W\left (\frac {b e^{\frac {c_1}{a}} x^{\frac {c}{a}}}{a}\right )}{b}\right \}\right \}\]
Maple ✓
cpu = 0.206 (sec), leaf count = 40
\[\left [y \left (x \right ) = {\mathrm e}^{\frac {c \textit {\_C1} -a \LambertW \left (\frac {b \,x^{\frac {c}{a}} {\mathrm e}^{\frac {c \textit {\_C1}}{a}}}{a}\right )+c \ln \left (x \right )}{a}}\right ]\] Mathematica raw input
DSolve[x*(a + b*y[x])*y'[x] == c*y[x],y[x],x]
Mathematica raw output
{{y[x] -> (a*ProductLog[(b*E^(C[1]/a)*x^(c/a))/a])/b}}
Maple raw input
dsolve(x*(a+b*y(x))*diff(y(x),x) = c*y(x), y(x))
Maple raw output
[y(x) = exp((c*_C1-a*LambertW(b/a*x^(c/a)*exp(c/a*_C1))+c*ln(x))/a)]