4.12.13 $x{y}^{\prime }(x)(a+by(x))=cy(x)$

ODE
$$x{y}^{\prime }(x)(a+by(x))=cy(x)$$ ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.0140668 (sec), leaf count = 31

$$\left\{\{y(x)\to \frac{aW\left(\frac{b{e}^{\frac{c_1}{a}}{x}^{\frac{c}{a}}}{a}\right)}{b}\}\right\}$$

Maple ✓
cpu = 0.008 (sec), leaf count = 24

$$\{\ln \left(x\right)-\frac{a\ln \left(y\left(x\right)\right)}{c}-\frac{by\left(x\right)}{c}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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