##### 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 = 0.307301 (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 = 0.403 (sec), leaf count = 94

$\left [y \left (x \right ) = \RootOf \left (\textit {\_Z}^{2}-{\mathrm e}^{\RootOf \left (x^{2} \left (-\left (\tanh ^{2}\left (\frac {\sqrt {b^{2}-4 a}\, \left (2 \textit {\_C1} +\textit {\_Z} +2 \ln \left (x \right )\right )}{2 b}\right )\right ) b^{2}+4 \left (\tanh ^{2}\left (\frac {\sqrt {b^{2}-4 a}\, \left (2 \textit {\_C1} +\textit {\_Z} +2 \ln \left (x \right )\right )}{2 b}\right )\right ) a +b^{2}+4 \,{\mathrm e}^{\textit {\_Z}}-4 a \right )\right )}+a +\textit {\_Z} b \right ) x\right ]$ 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))

Maple raw output

[y(x) = RootOf(_Z^2-exp(RootOf(x^2*(-tanh(1/2*(b^2-4*a)^(1/2)*(2*_C1+_Z+2*ln(x))
/b)^2*b^2+4*tanh(1/2*(b^2-4*a)^(1/2)*(2*_C1+_Z+2*ln(x))/b)^2*a+b^2+4*exp(_Z)-4*a
)))+a+_Z*b)*x]