Internal problem ID [9620]
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: 37.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {y^{\prime } x -y^{2} x -y a -b \,x^{n}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 216
dsolve(x*diff(y(x),x)=x*y(x)^2+a*y(x)+b*x^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} x^{\frac {n}{2}+\frac {1}{2}} \sqrt {b}\, \operatorname {BesselY}\left (-\frac {a -n}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{1+n}\right )}{\left (\operatorname {BesselY}\left (-\frac {a +1}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{1+n}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {a +1}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{1+n}\right )\right ) x}+\frac {\operatorname {BesselJ}\left (-\frac {a -n}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{1+n}\right ) \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{\left (\operatorname {BesselY}\left (-\frac {a +1}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{1+n}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {a +1}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{1+n}\right )\right ) x} \]
✓ Solution by Mathematica
Time used: 0.736 (sec). Leaf size: 855
DSolve[x*y'[x]==x*y[x]^2+a*y[x]+b*x^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}} \operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a-n}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )-\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}} \operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a+n+2}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+a \operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+\operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )-\sqrt {b} c_1 \left (x^n\right )^{\frac {n+1}{2 n}} \operatorname {Gamma}\left (\frac {n-a}{n+1}\right ) \operatorname {BesselJ}\left (\frac {n-a}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+\sqrt {b} c_1 \left (x^n\right )^{\frac {n+1}{2 n}} \operatorname {Gamma}\left (\frac {n-a}{n+1}\right ) \operatorname {BesselJ}\left (-\frac {a+n+2}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+(a+1) c_1 \operatorname {Gamma}\left (\frac {n-a}{n+1}\right ) \operatorname {BesselJ}\left (-\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )}{2 x \left (\operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+c_1 \operatorname {Gamma}\left (\frac {n-a}{n+1}\right ) \operatorname {BesselJ}\left (-\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )\right )} \\ y(x)\to -\frac {\frac {\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}} \left (\operatorname {BesselJ}\left (-\frac {a+n+2}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )-\operatorname {BesselJ}\left (\frac {n-a}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )\right )}{\operatorname {BesselJ}\left (-\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )}+a+1}{2 x} \\ y(x)\to -\frac {\frac {\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}} \left (\operatorname {BesselJ}\left (-\frac {a+n+2}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )-\operatorname {BesselJ}\left (\frac {n-a}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )\right )}{\operatorname {BesselJ}\left (-\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )}+a+1}{2 x} \\ \end{align*}