Internal problem ID [9587]
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]]
\[ \boxed {y^{\prime }-y^{2} a -b \,x^{n}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 207
dsolve(diff(y(x),x)=a*y(x)^2+b*x^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {b a}\, x^{\frac {n}{2}+1} \operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {b a}\, x^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {b a}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {b a}\, x^{\frac {n}{2}+1}-c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {b a}\, x^{\frac {n}{2}+1}}{n +2}\right )-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {b a}\, x^{\frac {n}{2}+1}}{n +2}\right )}{x a \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {b a}\, x^{\frac {n}{2}+1}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {b a}\, x^{\frac {n}{2}+1}}{n +2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.416 (sec). Leaf size: 433
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 \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \left (\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )\right )-c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )}{2 a x \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\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} \\ 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*}