4.7.22 $x(a+x)y^{\prime }(x)=y(x)(b+cy(x))$

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica ✓
cpu = 0.043622 (sec), leaf count = 50

$$\left\{\{y(x)\to -\frac{b{e}^{b{c}_1}{x}^{\frac{b}{a}}}{c{e}^{b{c}_1}{x}^{\frac{b}{a}}-(a+x{)}^{\frac{b}{a}}}\}\right\}$$

Maple ✓
cpu = 0.022 (sec), leaf count = 33

$$\{\frac{c}{b}-{(a+x)}^{\frac{b}{a}}{x}^{-\frac{b}{a}}\mathit{\_}\mathit{C}\mathit{1}+{\left(y\left(x\right)\right)}^{-1}=0\}$$ Mathematica raw input

DSolve[x*(a + x)*y'[x] == y[x]*(b + c*y[x]),y[x],x]

Mathematica raw output

{{y[x] -> -((b*E^(b*C[1])*x^(b/a))/(c*E^(b*C[1])*x^(b/a) - (a + x)^(b/a)))}}

Maple raw input

dsolve(x*(a+x)*diff(y(x),x) = (b+c*y(x))*y(x), y(x),'implicit')

Maple raw output

1/b*c-(a+x)^(b/a)*x^(-b/a)*_C1+1/y(x) = 0