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 = 0.00663421 (sec), leaf count = 133

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

Maple
cpu = 0.081 (sec), leaf count = 59

\[ \left \{ y \relax (x ) ={1\sqrt {a} \left ({{\sl J}_{0}\left (2\,\sqrt {a}\sqrt {x}\right )}{\it \_C1}+{{\sl Y}_{0}\left (2\,\sqrt {a}\sqrt {x}\right )} \right ) {\frac {1}{\sqrt {x}}} \left ({\it \_C1}\,{{\sl J}_{1}\left (2\,\sqrt {a}\sqrt {x}\right )}+{{\sl Y}_{1}\left (2\,\sqrt {a}\sqrt {x}\right )} \right ) ^{-1}} \right \} \] 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),'implicit')

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)))