Internal problem ID [9609]
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: 26.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-a \,x^{n} y-b \,x^{n -1}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 376
dsolve(diff(y(x),x)=y(x)^2+a*x^n*y(x)+b*x^(n-1),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (c_{1} a n +c_{1} a \right ) \operatorname {KummerU}\left (-\frac {a n +b}{a \left (1+n \right )}, \frac {n +2}{1+n}, \frac {a \,x^{1+n}}{1+n}\right )}{\left (\operatorname {KummerU}\left (\frac {a -b}{a \left (1+n \right )}, \frac {n +2}{1+n}, \frac {a \,x^{1+n}}{1+n}\right ) c_{1} +\operatorname {KummerM}\left (\frac {a -b}{a \left (1+n \right )}, \frac {n +2}{1+n}, \frac {a \,x^{1+n}}{1+n}\right )\right ) a x}+\frac {\left (-x^{1+n} c_{1} a^{2}+c_{1} a n +c_{1} b \right ) \operatorname {KummerU}\left (\frac {a -b}{a \left (1+n \right )}, \frac {n +2}{1+n}, \frac {a \,x^{1+n}}{1+n}\right )+\left (-a n -a -b \right ) \operatorname {KummerM}\left (-\frac {a n +b}{a \left (1+n \right )}, \frac {n +2}{1+n}, \frac {a \,x^{1+n}}{1+n}\right )+\left (-a^{2} x^{1+n}+a n +b \right ) \operatorname {KummerM}\left (\frac {a -b}{a \left (1+n \right )}, \frac {n +2}{1+n}, \frac {a \,x^{1+n}}{1+n}\right )}{\left (\operatorname {KummerU}\left (\frac {a -b}{a \left (1+n \right )}, \frac {n +2}{1+n}, \frac {a \,x^{1+n}}{1+n}\right ) c_{1} +\operatorname {KummerM}\left (\frac {a -b}{a \left (1+n \right )}, \frac {n +2}{1+n}, \frac {a \,x^{1+n}}{1+n}\right )\right ) a x} \]
✓ Solution by Mathematica
Time used: 0.532 (sec). Leaf size: 427
DSolve[y'[x]==y[x]^2+a*x^n*y[x]+b*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (x^n\right )^{\frac {1}{n}} \left (e^{\frac {i \pi }{n+1}} n a^{\frac {1}{n+1}} \left (\frac {\left (x^n\right )^{\frac {1}{n}+1} (a n+a+b) \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},2+\frac {1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )}{n+2}-\left (a \left (x^n\right )^{\frac {1}{n}+1}+1\right ) \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},1+\frac {1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )+b c_1 \left (\frac {1}{n}+1\right )^{\frac {1}{n+1}} n^{\frac {1}{n+1}} x^n \operatorname {Hypergeometric1F1}\left (1-\frac {b}{n a+a},2-\frac {1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )}{n x \left ((-1)^{\frac {1}{n+1}} a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},1+\frac {1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+c_1 \left (\frac {1}{n}+1\right )^{\frac {1}{n+1}} n^{\frac {1}{n+1}} \operatorname {Hypergeometric1F1}\left (-\frac {b}{n a+a},\frac {n}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )} \\ y(x)\to \frac {b x^{n-1} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (1-\frac {b}{n a+a},2-\frac {1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )}{n \operatorname {Hypergeometric1F1}\left (-\frac {b}{n a+a},\frac {n}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )} \\ \end{align*}