4.2.4 \(y'(x)=3 \left (a+b y(x)^2+b x\right )\)

ODE
\[ y'(x)=3 \left (a+b y(x)^2+b x\right ) \] ODE Classification

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.0248185 (sec), leaf count = 123

\[\left \{\left \{y(x)\to \frac {b \left (c_1 \text {Ai}'\left (-\frac {3^{2/3} b (a+b x)}{\left (-b^2\right )^{2/3}}\right )+\text {Bi}'\left (-\frac {3^{2/3} b (a+b x)}{\left (-b^2\right )^{2/3}}\right )\right )}{\sqrt [3]{3} \left (-b^2\right )^{2/3} \left (c_1 \text {Ai}\left (-\frac {3^{2/3} b (a+b x)}{\left (-b^2\right )^{2/3}}\right )+\text {Bi}\left (-\frac {3^{2/3} b (a+b x)}{\left (-b^2\right )^{2/3}}\right )\right )}\right \}\right \}\]

Maple
cpu = 0.284 (sec), leaf count = 80

\[ \left \{ y \relax (x ) ={{3}^{{\frac {2}{3}}} \left ({{\rm Ai}^{(1)}\left (-{{3}^{{\frac {2}{3}}} \left (bx+a \right ) {\frac {1}{\sqrt [3]{b}}}}\right )}{\it \_C1}+{{\rm Bi}^{(1)}\left (-{{3}^{{\frac {2}{3}}} \left (bx+a \right ) {\frac {1}{\sqrt [3]{b}}}}\right )} \right ) {\frac {1}{\sqrt [3]{b}}} \left (3\,{{\rm Ai}\left (-{\frac {{3}^{2/3} \left (bx+a \right ) }{\sqrt [3]{b}}}\right )}{\it \_C1}+3\,{{\rm Bi}\left (-{\frac {{3}^{2/3} \left (bx+a \right ) }{\sqrt [3]{b}}}\right )} \right ) ^{-1}} \right \} \] Mathematica raw input

DSolve[y'[x] == 3*(a + b*x + b*y[x]^2),y[x],x]

Mathematica raw output

{{y[x] -> (b*(AiryBiPrime[-((3^(2/3)*b*(a + b*x))/(-b^2)^(2/3))] + AiryAiPrime[-
((3^(2/3)*b*(a + b*x))/(-b^2)^(2/3))]*C[1]))/(3^(1/3)*(-b^2)^(2/3)*(AiryBi[-((3^
(2/3)*b*(a + b*x))/(-b^2)^(2/3))] + AiryAi[-((3^(2/3)*b*(a + b*x))/(-b^2)^(2/3))
]*C[1]))}}

Maple raw input

dsolve(diff(y(x),x) = 3*a+3*b*x+3*b*y(x)^2, y(x),'implicit')

Maple raw output

y(x) = (AiryAi(1,-3^(2/3)*(b*x+a)/b^(1/3))*_C1+AiryBi(1,-3^(2/3)*(b*x+a)/b^(1/3)
))/b^(1/3)*3^(2/3)/(3*AiryAi(-3^(2/3)*(b*x+a)/b^(1/3))*_C1+3*AiryBi(-3^(2/3)*(b*
x+a)/b^(1/3)))