##### 4.2.9 $$y'(x)=a x^2+b y(x)^2$$

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

[[_Riccati, _special]]

Book solution method
Riccati ODE, Main form

Mathematica
cpu = 0.227735 (sec), leaf count = 175

$\left \{\left \{y(x)\to \frac {\sqrt {a} \sqrt {b} x^2 \left (-2 J_{-\frac {3}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 \left (J_{\frac {3}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )-J_{-\frac {5}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )\right )-c_1 J_{-\frac {1}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )}{2 b x \left (J_{\frac {1}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )+c_1 J_{-\frac {1}{4}}\left (\frac {1}{2} \sqrt {a} \sqrt {b} x^2\right )\right )}\right \}\right \}$

Maple
cpu = 0.129 (sec), leaf count = 71

$\left [y \left (x \right ) = -\frac {\sqrt {a b}\, x \left (\BesselJ \left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right ) \textit {\_C1} +\BesselY \left (-\frac {3}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )}{b \left (\textit {\_C1} \BesselJ \left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )+\BesselY \left (\frac {1}{4}, \frac {\sqrt {a b}\, x^{2}}{2}\right )\right )}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-(BesselJ[-1/4, (Sqrt[a]*Sqrt[b]*x^2)/2]*C[1]) + Sqrt[a]*Sqrt[b]*x^2*
(-2*BesselJ[-3/4, (Sqrt[a]*Sqrt[b]*x^2)/2] + (-BesselJ[-5/4, (Sqrt[a]*Sqrt[b]*x^
2)/2] + BesselJ[3/4, (Sqrt[a]*Sqrt[b]*x^2)/2])*C[1]))/(2*b*x*(BesselJ[1/4, (Sqrt
[a]*Sqrt[b]*x^2)/2] + BesselJ[-1/4, (Sqrt[a]*Sqrt[b]*x^2)/2]*C[1]))}}

Maple raw input

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

Maple raw output

[y(x) = -(a*b)^(1/2)*x*(BesselJ(-3/4,1/2*(a*b)^(1/2)*x^2)*_C1+BesselY(-3/4,1/2*(
a*b)^(1/2)*x^2))/b/(_C1*BesselJ(1/4,1/2*(a*b)^(1/2)*x^2)+BesselY(1/4,1/2*(a*b)^(
1/2)*x^2))]