Internal problem ID [9584]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2} a -b x -c=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 85
dsolve(diff(y(x),x)=a*y(x)^2+b*x+c,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {b}{\sqrt {a}}\right )^{\frac {1}{3}} \left (\operatorname {AiryAi}\left (1, -\frac {x b +c}{\left (\frac {b}{\sqrt {a}}\right )^{\frac {2}{3}}}\right ) c_{1} +\operatorname {AiryBi}\left (1, -\frac {x b +c}{\left (\frac {b}{\sqrt {a}}\right )^{\frac {2}{3}}}\right )\right )}{\sqrt {a}\, \left (c_{1} \operatorname {AiryAi}\left (-\frac {x b +c}{\left (\frac {b}{\sqrt {a}}\right )^{\frac {2}{3}}}\right )+\operatorname {AiryBi}\left (-\frac {x b +c}{\left (\frac {b}{\sqrt {a}}\right )^{\frac {2}{3}}}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.188 (sec). Leaf size: 143
DSolve[y'[x]==a*y[x]^2+b*x+c,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b \left (\operatorname {AiryBiPrime}\left (-\frac {a (c+b x)}{(-a b)^{2/3}}\right )+c_1 \operatorname {AiryAiPrime}\left (-\frac {a (c+b x)}{(-a b)^{2/3}}\right )\right )}{(-a b)^{2/3} \left (\operatorname {AiryBi}\left (-\frac {a (c+b x)}{(-a b)^{2/3}}\right )+c_1 \operatorname {AiryAi}\left (-\frac {a (c+b x)}{(-a b)^{2/3}}\right )\right )} \\ y(x)\to \frac {b \operatorname {AiryAiPrime}\left (-\frac {a (c+b x)}{(-a b)^{2/3}}\right )}{(-a b)^{2/3} \operatorname {AiryAi}\left (-\frac {a (c+b x)}{(-a b)^{2/3}}\right )} \\ \end{align*}