Internal problem ID [10344]
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]]
\[ \boxed {y^{\prime }-a y^{2}=b \,x^{n}} \]
✓ 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: 0.696 (sec). Leaf size: 605
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} \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 )-\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+\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 )-\sqrt {a} \sqrt {b} c_1 x^{\frac {n}{2}+1} \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+\sqrt {a} \sqrt {b} c_1 x^{\frac {n}{2}+1} \operatorname {Gamma}\left (\frac {n+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 )+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 {\sqrt {a} \sqrt {b} x^{n/2} \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 )}{\operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )}-\frac {1}{x}}{2 a} \end{align*}