4.2.7 \(y'(x)=a+b x+c y(x)^2\)

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

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.0220407 (sec), leaf count = 93

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

Maple
cpu = 0.107 (sec), leaf count = 85

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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