4.12.13 \(x y'(x) (a+b y(x))=c y(x)\)

ODE
\[ x y'(x) (a+b y(x))=c y(x) \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.0140668 (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 = 0.008 (sec), leaf count = 24

\[ \left \{ \ln \relax (x ) -{\frac {a\ln \left (y \relax (x ) \right ) }{c}}-{\frac {by \relax (x ) }{c}}+{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

ln(x)-1/c*a*ln(y(x))-b/c*y(x)+_C1 = 0