Internal problem ID [9611]
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]
Solve \begin {gather*} \boxed {y^{\prime }-a y^{2}-b y-x c -k=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 194
dsolve(diff(y(x),x)=a*y(x)^2+b*y(x)+c*x+k,y(x), singsol=all)
\[ y \relax (x ) = \frac {2 \left (\frac {c}{\sqrt {a}}\right )^{\frac {1}{3}} \sqrt {a}\, \left (\AiryAi \left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right ) c_{1}+\AiryBi \left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )\right )-b \left (c_{1} \AiryAi \left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )+\AiryBi \left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )\right )}{2 a \left (c_{1} \AiryAi \left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )+\AiryBi \left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.5 (sec). Leaf size: 339
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 \text {Bi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+b c_1 \text {Ai}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+2 \sqrt [3]{-a c} \left (\text {Bi}'\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+c_1 \text {Ai}'\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )\right )}{2 a \left (\text {Bi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+c_1 \text {Ai}\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} \text {Ai}'\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}{\text {Ai}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}+b}{2 a} \\ y(x)\to -\frac {\frac {2 \sqrt [3]{-a c} \text {Ai}'\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}{\text {Ai}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}+b}{2 a} \\ \end{align*}