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 = 0.00597792 (sec), leaf count = 175

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

Maple
cpu = 0.092 (sec), leaf count = 71

\[ \left \{ y \relax (x ) =-{\frac {x}{b}\sqrt {ab} \left ({{\sl J}_{-{\frac {3}{4}}}\left ({\frac {{x}^{2}}{2}\sqrt {ab}}\right )}{\it \_C1}+{{\sl Y}_{-{\frac {3}{4}}}\left ({\frac {{x}^{2}}{2}\sqrt {ab}}\right )} \right ) \left ({\it \_C1}\,{{\sl J}_{{\frac {1}{4}}}\left ({\frac {{x}^{2}}{2}\sqrt {ab}}\right )}+{{\sl Y}_{{\frac {1}{4}}}\left ({\frac {{x}^{2}}{2}\sqrt {ab}}\right )} \right ) ^{-1}} \right \} \] 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),'implicit')

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))