Internal problem ID [10375]
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: 35.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {y^{\prime } x -a y^{2}-b y=c \,x^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 225
dsolve(x*diff(y(x),x)=a*y(x)^2+b*y(x)+c*x^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {a c}\, x^{\frac {n}{2}} c_{1} \operatorname {BesselY}\left (\frac {b +n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )}{a \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )}+\frac {\operatorname {BesselJ}\left (\frac {b +n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) \sqrt {a c}\, x^{\frac {n}{2}}-\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} b -b \operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )}{a \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.513 (sec). Leaf size: 402
DSolve[x*y'[x]==a*y[x]^2+b*y[x]+c*x^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {c} x^{n/2} \left (-2 \operatorname {BesselJ}\left (\frac {b}{n}-1,\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+c_1 \left (\operatorname {BesselJ}\left (1-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )-\operatorname {BesselJ}\left (-\frac {b+n}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )\right )\right )-b c_1 \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )}{2 a \left (\operatorname {BesselJ}\left (\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+c_1 \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )\right )} y(x)\to -\frac {-\sqrt {a} \sqrt {c} x^{n/2} \operatorname {BesselJ}\left (1-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+\sqrt {a} \sqrt {c} x^{n/2} \operatorname {BesselJ}\left (-\frac {b+n}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+b \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )}{2 a \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )} \end{align*}