4.11.28 \(x (a+y(x)) y'(x)=y(x) (A+B x)\)

ODE
\[ x (a+y(x)) y'(x)=y(x) (A+B x) \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.0177155 (sec), leaf count = 31

\[\left \{\left \{y(x)\to a W\left (\frac {x^{\frac {A}{a}} e^{\frac {B x+c_1}{a}}}{a}\right )\right \}\right \}\]

Maple
cpu = 0.01 (sec), leaf count = 22

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

DSolve[x*(a + y[x])*y'[x] == (A + B*x)*y[x],y[x],x]

Mathematica raw output

{{y[x] -> a*ProductLog[(E^((B*x + C[1])/a)*x^(A/a))/a]}}

Maple raw input

dsolve(x*(a+y(x))*diff(y(x),x) = y(x)*(B*x+A), y(x),'implicit')

Maple raw output

B*x+A*ln(x)-y(x)-a*ln(y(x))+_C1 = 0