Internal problem ID [9591]
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: 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Riccati, _special]]
Solve \begin {gather*} \boxed {y^{\prime }-a y^{2}-b \,x^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 207
dsolve(diff(y(x),x)=a*y(x)^2+b*x^n,y(x), singsol=all)
\[ y \relax (x ) = \frac {\BesselJ \left (\frac {3+n}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a b}\, x^{\frac {n}{2}+1} c_{1}+\BesselY \left (\frac {3+n}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a b}\, x^{\frac {n}{2}+1}-c_{1} \BesselJ \left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )-\BesselY \left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )}{x a \left (c_{1} \BesselJ \left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )+\BesselY \left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.261 (sec). Leaf size: 324
DSolve[y'[x]==a*y[x]^2+b*x^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \left (-2 J_{\frac {1}{n+2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+c_1 \left (J_{\frac {n+1}{n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-J_{-\frac {n+3}{n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )\right )-c_1 J_{-\frac {1}{n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )}{2 a x \left (J_{\frac {1}{n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+c_1 J_{-\frac {1}{n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )} \\ y(x)\to \frac {\frac {\, _0F_1\left (;-\frac {1}{n+2};-\frac {a b x^{n+2}}{(n+2)^2}\right )}{\, _0F_1\left (;\frac {n+1}{n+2};-\frac {a b x^{n+2}}{(n+2)^2}\right )}-1}{a x} \\ \end{align*}