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

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

[_separable]

Book solution method
Separable ODE, Neither variable missing

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

\[\left \{\left \{y(x)\to -\frac {b e^{b c_1} x^{\frac {b}{a}}}{c e^{b c_1} x^{\frac {b}{a}}-(a+x)^{\frac {b}{a}}}\right \}\right \}\]

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

\[ \left \{ {\frac {c}{b}}- \left (a+x \right ) ^{{\frac {b}{a}}}{x}^{-{\frac {b}{a}}}{\it \_C1}+ \left (y \relax (x ) \right ) ^{-1}=0 \right \} \] 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