##### 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 = 0.246754 (sec), leaf count = 93

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

Maple
cpu = 0.087 (sec), leaf count = 85

$\left [y \left (x \right ) = \frac {\left (\frac {b}{\sqrt {c}}\right )^{\frac {1}{3}} \left (\AiryAi \left (1, -\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{\frac {2}{3}}}\right ) \textit {\_C1} +\AiryBi \left (1, -\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{\frac {2}{3}}}\right )\right )}{\sqrt {c}\, \left (\textit {\_C1} \AiryAi \left (-\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{\frac {2}{3}}}\right )+\AiryBi \left (-\frac {b x +a}{\left (\frac {b}{\sqrt {c}}\right )^{\frac {2}{3}}}\right )\right )}\right ]$ 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))

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