4.11.3 \(y'(x) (a x+b y(x))+y(x)=0\)

ODE
\[ y'(x) (a x+b y(x))+y(x)=0 \] ODE Classification

[[_homogeneous, `class A`], _rational, [_Abel, `2nd type``class A`]]

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.0389605 (sec), leaf count = 36

\[\text {Solve}\left [\frac {\log \left (a+\frac {b y(x)}{x}+1\right )+a \log \left (\frac {y(x)}{x}\right )}{a+1}+\log (x)=c_1,y(x)\right ]\]

Maple
cpu = 0.023 (sec), leaf count = 23

\[ \left \{ x+{\frac {by \relax (x ) }{1+a}}- \left (y \relax (x ) \right ) ^{-a}{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y[x] + (a*x + b*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[Log[x] + (a*Log[y[x]/x] + Log[1 + a + (b*y[x])/x])/(1 + a) == C[1], y[x]]

Maple raw input

dsolve((a*x+b*y(x))*diff(y(x),x)+y(x) = 0, y(x),'implicit')

Maple raw output

x+b/(1+a)*y(x)-y(x)^(-a)*_C1 = 0