61.2.1 problem 1

Internal problem ID [12007]
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
Date solved : Monday, January 27, 2025 at 11:48:10 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a y^{2}+b x +c \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 85

dsolve(diff(y(x),x)=a*y(x)^2+b*x+c,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.207 (sec). Leaf size: 143

DSolve[D[y[x],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*}