Internal problem ID [10356]
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]
\[ \boxed {y^{\prime }-y^{2}-a \,x^{n} y=b \,x^{n -1}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 267
dsolve(diff(y(x),x)=y(x)^2+a*x^n*y(x)+b*x^(n-1),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-a \left (n +1\right ) \left (a -b \right ) \operatorname {KummerM}\left (\frac {a \left (n +2\right )-b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right )+\left (\left (a -b \right ) c_{1} \operatorname {KummerU}\left (\frac {2+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right )-a \left (\operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right ) c_{1} +\operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right )\right ) \left (n +1\right )\right ) b}{a^{2} \left (n +1\right ) x \left (\operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right ) c_{1} +\operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right )\right )} \]
✓ Solution by Mathematica
Time used: 0.956 (sec). Leaf size: 453
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 (-(-1)^{\frac {1}{n+1}} n (n+2) a^{\frac {1}{n+1}} \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},\frac {n+2}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+x^n \left (-(-1)^{\frac {1}{n+1}} n (a-b) a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {n a+2 a-b}{n a+a},\frac {2 n+3}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+b c_1 \left (\frac {1}{n}+1\right )^{\frac {1}{n+1}} n^{\frac {1}{n+1}} (n+2) \operatorname {Hypergeometric1F1}\left (\frac {n a+a-b}{n a+a},\frac {2 n+1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )\right )}{n (n+2) 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},\frac {n+2}{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 (\frac {n a+a-b}{n a+a},\frac {2 n+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*}