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

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.138588 (sec), leaf count = 71

\[\text {Solve}\left [\frac {1}{2} \log \left (a+\frac {b y(x)}{x}+\frac {y(x)^2}{x^2}\right )+\log (x)=\frac {b \tan ^{-1}\left (\frac {b+\frac {2 y(x)}{x}}{\sqrt {4 a-b^2}}\right )}{\sqrt {4 a-b^2}}+c_1,y(x)\right ]\]

Maple
cpu = 0.016 (sec), leaf count = 67

\[ \left \{ -{\frac {1}{2}\ln \left ({\frac {a{x}^{2}+bxy \relax (x ) + \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) }-{b{\it Artanh} \left ({\frac {bx+2\,y \relax (x ) }{x}{\frac {1}{\sqrt {{b}^{2}-4\,a}}}} \right ) {\frac {1}{\sqrt {{b}^{2}-4\,a}}}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

Solve[Log[x] + Log[a + (b*y[x])/x + y[x]^2/x^2]/2 == (b*ArcTan[(b + (2*y[x])/x)/
Sqrt[4*a - b^2]])/Sqrt[4*a - b^2] + C[1], y[x]]

Maple raw input

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

Maple raw output

-1/2*ln((a*x^2+b*x*y(x)+y(x)^2)/x^2)-b/(b^2-4*a)^(1/2)*arctanh((b*x+2*y(x))/x/(b
^2-4*a)^(1/2))-ln(x)-_C1 = 0