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 = 0.00577601 (sec), leaf count = 189

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

Maple ✓
cpu = 0.077 (sec), leaf count = 59

\[ \left \{ y \left ( x \right ) ={\frac {1}{b}\sqrt [3]{ab} \left ( {{\rm Ai}^{(1)}\left (-\sqrt [3]{ab}x\right )}{\it \_C1}+{{\rm Bi}^{(1)}\left (-\sqrt [3]{ab}x\right )} \right ) \left ( {\it \_C1}\,{{\rm Ai}\left (-\sqrt [3]{ab}x\right )}+{{\rm Bi}\left (-\sqrt [3]{ab}x\right )} \right ) ^{-1}} \right \} \] 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),'implicit')

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/(_C
1*AiryAi(-(a*b)^(1/3)*x)+AiryBi(-(a*b)^(1/3)*x))