4.2.6 $$y'(x)=a x+b y(x)^2$$

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

[[_Riccati, _special]]

Book solution method
Riccati ODE, Main form

Mathematica
cpu = 0.218748 (sec), leaf count = 189

$\left \{\left \{y(x)\to \frac {\sqrt {a} \sqrt {b} x^{3/2} \left (-2 J_{-\frac {2}{3}}\left (\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )+c_1 \left (J_{\frac {2}{3}}\left (\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )-J_{-\frac {4}{3}}\left (\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )\right )\right )-c_1 J_{-\frac {1}{3}}\left (\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )}{2 b x \left (J_{\frac {1}{3}}\left (\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )+c_1 J_{-\frac {1}{3}}\left (\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )\right )}\right \}\right \}$

Maple
cpu = 0.119 (sec), leaf count = 59

$\left [y \left (x \right ) = \frac {\left (a b \right )^{\frac {1}{3}} \left (\AiryAi \left (1, -\left (a b \right )^{\frac {1}{3}} x \right ) \textit {\_C1} +\AiryBi \left (1, -\left (a b \right )^{\frac {1}{3}} x \right )\right )}{b \left (\textit {\_C1} \AiryAi \left (-\left (a b \right )^{\frac {1}{3}} x \right )+\AiryBi \left (-\left (a b \right )^{\frac {1}{3}} x \right )\right )}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-(BesselJ[-1/3, (2*Sqrt[a]*Sqrt[b]*x^(3/2))/3]*C[1]) + Sqrt[a]*Sqrt[b
]*x^(3/2)*(-2*BesselJ[-2/3, (2*Sqrt[a]*Sqrt[b]*x^(3/2))/3] + (-BesselJ[-4/3, (2*
Sqrt[a]*Sqrt[b]*x^(3/2))/3] + BesselJ[2/3, (2*Sqrt[a]*Sqrt[b]*x^(3/2))/3])*C[1])
)/(2*b*x*(BesselJ[1/3, (2*Sqrt[a]*Sqrt[b]*x^(3/2))/3] + BesselJ[-1/3, (2*Sqrt[a]
*Sqrt[b]*x^(3/2))/3]*C[1]))}}

Maple raw input

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

Maple raw output

[y(x) = (a*b)^(1/3)*(AiryAi(1,-(a*b)^(1/3)*x)*_C1+AiryBi(1,-(a*b)^(1/3)*x))/b/(_
C1*AiryAi(-(a*b)^(1/3)*x)+AiryBi(-(a*b)^(1/3)*x))]