4.4.31 \(y(x) (a x y(x)+2)+b x+x y'(x)=0\)

ODE
\[ y(x) (a x y(x)+2)+b x+x y'(x)=0 \] ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.0148677 (sec), leaf count = 43

\[\left \{\left \{y(x)\to -\sqrt {\frac {b}{a}} \tan \left (a x \sqrt {\frac {b}{a}}-c_1\right )-\frac {1}{a x}\right \}\right \}\]

Maple
cpu = 0.072 (sec), leaf count = 64

\[ \left \{ -ay \relax (x ) +{\frac {1}{x} \left (i\sqrt {a}\sqrt {b}x-1 \right ) }+2\,{{{\rm e}^{-2\,ix\sqrt {a}\sqrt {b}}} \left ({\frac {i{{\rm e}^{-2\,ix\sqrt {a}\sqrt {b}}}}{\sqrt {a}\sqrt {b}}}-2\,{\it \_C1} \right ) ^{-1}}=0 \right \} \] Mathematica raw input

DSolve[b*x + y[x]*(2 + a*x*y[x]) + x*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(1/(a*x)) - Sqrt[b/a]*Tan[a*Sqrt[b/a]*x - C[1]]}}

Maple raw input

dsolve(x*diff(y(x),x)+b*x+(2+a*x*y(x))*y(x) = 0, y(x),'implicit')

Maple raw output

-a*y(x)+(I*a^(1/2)*b^(1/2)*x-1)/x+2*exp(-2*I*x*a^(1/2)*b^(1/2))/(I/a^(1/2)/b^(1/
2)*exp(-2*I*x*a^(1/2)*b^(1/2))-2*_C1) = 0