Internal problem ID [9607]
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: 24.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2} a -b y-c x -k=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 194
dsolve(diff(y(x),x)=a*y(x)^2+b*y(x)+c*x+k,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 \sqrt {a}\, \left (\frac {c}{\sqrt {a}}\right )^{\frac {1}{3}} \left (\operatorname {AiryAi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right ) c_{1} +\operatorname {AiryBi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )\right )-\left (c_{1} \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )\right ) b}{2 a \left (c_{1} \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.303 (sec). Leaf size: 264
DSolve[y'[x]==a*y[x]^2+b*y[x]+c*x+k,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {b \operatorname {AiryBi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+b c_1 \operatorname {AiryAi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+2 \sqrt [3]{-a c} \left (\operatorname {AiryBiPrime}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+c_1 \operatorname {AiryAiPrime}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )\right )}{2 a \left (\operatorname {AiryBi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+c_1 \operatorname {AiryAi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )\right )} \\ y(x)\to -\frac {\frac {2 \sqrt [3]{-a c} \operatorname {AiryAiPrime}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}{\operatorname {AiryAi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}+b}{2 a} \\ \end{align*}