##### 4.4.24 $$a+x y'(x)+x y(x)^2=0$$

ODE
$a+x y'(x)+x y(x)^2=0$ ODE Classiﬁcation

[_rational, [_Riccati, _special]]

Book solution method
Riccati ODE, Main form

Mathematica
cpu = 0.247517 (sec), leaf count = 133

$\left \{\left \{y(x)\to \frac {i \sqrt {-a} (-2+c_1) \sqrt {x} J_0\left (2 i \sqrt {-a} \sqrt {x}\right )+c_1 \left (J_1\left (2 i \sqrt {-a} \sqrt {x}\right )-i \sqrt {-a} \sqrt {x} J_2\left (2 i \sqrt {-a} \sqrt {x}\right )\right )}{2 (-1+c_1) x J_1\left (2 i \sqrt {-a} \sqrt {x}\right )}\right \}\right \}$

Maple
cpu = 0.118 (sec), leaf count = 59

$\left [y \left (x \right ) = \frac {\sqrt {a}\, \left (\BesselJ \left (0, 2 \sqrt {a}\, \sqrt {x}\right ) \textit {\_C1} +\BesselY \left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )}{\sqrt {x}\, \left (\textit {\_C1} \BesselJ \left (1, 2 \sqrt {a}\, \sqrt {x}\right )+\BesselY \left (1, 2 \sqrt {a}\, \sqrt {x}\right )\right )}\right ]$ Mathematica raw input

DSolve[a + x*y[x]^2 + x*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (I*Sqrt[-a]*Sqrt[x]*BesselJ[0, (2*I)*Sqrt[-a]*Sqrt[x]]*(-2 + C[1]) + (
BesselJ[1, (2*I)*Sqrt[-a]*Sqrt[x]] - I*Sqrt[-a]*Sqrt[x]*BesselJ[2, (2*I)*Sqrt[-a
]*Sqrt[x]])*C[1])/(2*x*BesselJ[1, (2*I)*Sqrt[-a]*Sqrt[x]]*(-1 + C[1]))}}

Maple raw input

dsolve(x*diff(y(x),x)+a+x*y(x)^2 = 0, y(x))

Maple raw output

[y(x) = a^(1/2)*(BesselJ(0,2*a^(1/2)*x^(1/2))*_C1+BesselY(0,2*a^(1/2)*x^(1/2)))/
x^(1/2)/(_C1*BesselJ(1,2*a^(1/2)*x^(1/2))+BesselY(1,2*a^(1/2)*x^(1/2)))]