4.2.7 $y^{\prime }(x)=a+bx+cy(x{)}^2$

ODE
$$y^{\prime }(x)=a+bx+cy(x{)}^2$$ ODE Classification

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

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

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

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

$$\{y\left(x\right)=1\sqrt[3]{b\frac{1}{\sqrt{c}}}({\mathrm{A}\mathrm{i}}^{(1)}(-(bx+a){\left(b\frac{1}{\sqrt{c}}\right)}^{-\frac{2}{3}})\mathit{\_}\mathit{C}\mathit{1}+{\mathrm{B}\mathrm{i}}^{(1)}(-(bx+a){\left(b\frac{1}{\sqrt{c}}\right)}^{-\frac{2}{3}}))\frac{1}{\sqrt{c}}{(\mathrm{A}\mathrm{i}(-(bx+a){\left(b\frac{1}{\sqrt{c}}\right)}^{-\frac{2}{3}})\mathit{\_}\mathit{C}\mathit{1}+\mathrm{B}\mathrm{i}(-(bx+a){\left(b\frac{1}{\sqrt{c}}\right)}^{-\frac{2}{3}}))}^{-1}\}$$ 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)))