Internal problem ID [9605]
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: 22.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {\left (x^{n} a +b \right ) y^{\prime }-b y^{2}-a \,x^{n -2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 333
dsolve((a*x^n+b)*diff(y(x),x)=b*y(x)^2+a*x^(n-2),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (-x^{2 n} c_{1} a^{2} n -x^{n} c_{1} a b n \right ) \operatorname {hypergeom}\left (\left [2, \frac {1+n}{n}\right ], \left [\frac {2 n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right )}{\left (\operatorname {hypergeom}\left (\left [\frac {2}{n}\right ], \left [\right ], -\frac {a \,x^{n}}{b}\right ) x +\operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right ) c_{1} \right ) \left (n -1\right ) b^{2} x}-\frac {\left (x^{n} c_{1} a b n -x^{n} c_{1} a b \right ) \operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right )+\left (2 a b n \,x^{1+n}-2 a b \,x^{1+n}+b^{2} n x -b^{2} x \right ) \operatorname {hypergeom}\left (\left [\frac {2}{n}\right ], \left [\right ], -\frac {a \,x^{n}}{b}\right )+\left (-2 n \,x^{2 n +1} a^{2}+2 x^{2 n +1} a^{2}-2 a b n \,x^{1+n}+2 a b \,x^{1+n}\right ) \operatorname {hypergeom}\left (\left [\frac {n +2}{n}\right ], \left [\right ], -\frac {a \,x^{n}}{b}\right )}{\left (\operatorname {hypergeom}\left (\left [\frac {2}{n}\right ], \left [\right ], -\frac {a \,x^{n}}{b}\right ) x +\operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right ) c_{1} \right ) \left (n -1\right ) b^{2} x} \]
✓ Solution by Mathematica
Time used: 1.088 (sec). Leaf size: 250
DSolve[(a*x^n+b)*y'[x]==b*y[x]^2+a*x^(n-2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \left (a (n-1) x^n+b n\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},\frac {n-2}{n},\frac {n-1}{n},-\frac {a x^n}{b}\right )-b \left ((-1)^{\frac {1}{n}} \left (-\frac {a x^n}{b}\right )^{\frac {1}{n}}+c_1 n \operatorname {Hypergeometric2F1}\left (\frac {n-2}{n},-\frac {n+1}{n},\frac {n-1}{n},-\frac {a x^n}{b}\right )\right )}{b x \left ((-1)^{\frac {1}{n}} \left (-\frac {a x^n}{b}\right )^{\frac {1}{n}}+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},\frac {n-2}{n},\frac {n-1}{n},-\frac {a x^n}{b}\right )\right )} \\ y(x)\to \frac {a x^{n-1} \left (\frac {n \left (a x^n+b\right ) \operatorname {Hypergeometric2F1}\left (2,1+\frac {1}{n},2-\frac {1}{n},-\frac {a x^n}{b}\right )}{\operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},\frac {n-1}{n},-\frac {a x^n}{b}\right )}+b (-n)+b\right )}{b^2 (n-1)} \\ \end{align*}