Internal problem ID [10352]
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]
\[ \boxed {\left (x^{n} a +b \right ) y^{\prime }-b y^{2}=a \,x^{-2+n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 224
dsolve((a*x^n+b)*diff(y(x),x)=b*y(x)^2+a*x^(n-2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {a \,x^{n}+b}{b}\right )^{\frac {2}{n}} \left (a n c_{1} \left (a^{2} x^{3 n}+2 a b \,x^{2 n}+b^{2} x^{n}\right ) \operatorname {hypergeom}\left (\left [2, \frac {n +1}{n}\right ], \left [\frac {2 n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right )-\left (n -1\right ) b \left (a c_{1} \left (a \,x^{2 n}+b \,x^{n}\right ) \operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right )+\left (\frac {a \,x^{n}+b}{b}\right )^{-\frac {2}{n}} b \left (x^{n +1} a +b x \right )\right )\right )}{b^{2} \left (n -1\right ) x \left (a \,x^{n}+b \right ) \left (x +\operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right ) c_{1} \left (\frac {a \,x^{n}+b}{b}\right )^{\frac {2}{n}}\right )} \]
✓ Solution by Mathematica
Time used: 1.899 (sec). Leaf size: 289
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 {-b^2 (-1)^{\frac {1}{n}} (n-1) \left (-\frac {a x^n}{b}\right )^{\frac {1}{n}}-a b c_1 (n-1) x^n \left (\frac {a x^n}{b}+1\right )^{2/n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},\frac {n-1}{n},-\frac {a x^n}{b}\right )+a c_1 n x^n \left (a x^n+b\right ) \left (\frac {a x^n}{b}+1\right )^{2/n} \operatorname {Hypergeometric2F1}\left (2,1+\frac {1}{n},2-\frac {1}{n},-\frac {a x^n}{b}\right )}{b^2 (n-1) x \left ((-1)^{\frac {1}{n}} \left (-\frac {a x^n}{b}\right )^{\frac {1}{n}}+c_1 \left (\frac {a x^n}{b}+1\right )^{2/n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{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*}