Internal problem ID [9613]
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]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a \,x^{n} y-b \,x^{-1+n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 378
dsolve(diff(y(x),x)=y(x)^2+a*x^n*y(x)+b*x^(n-1),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-c_{1} a n -a c_{1}\right ) \KummerU \left (-\frac {n a +b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {x^{n +1} a}{n +1}\right )}{\left (\KummerU \left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {x^{n +1} a}{n +1}\right ) c_{1}+\KummerM \left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {x^{n +1} a}{n +1}\right )\right ) x a}-\frac {\left (x^{n +1} c_{1} a^{2}-c_{1} a n -b c_{1}\right ) \KummerU \left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {x^{n +1} a}{n +1}\right )+\left (n a +a +b \right ) \KummerM \left (-\frac {n a +b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {x^{n +1} a}{n +1}\right )+\left (x^{n +1} a^{2}-n a -b \right ) \KummerM \left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {x^{n +1} a}{n +1}\right )}{\left (\KummerU \left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {x^{n +1} a}{n +1}\right ) c_{1}+\KummerM \left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {x^{n +1} a}{n +1}\right )\right ) x a} \]
✓ Solution by Mathematica
Time used: 0.83 (sec). Leaf size: 522
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) \, _1F_1\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 ) \, _1F_1\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 \, _1F_1\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}} \, _1F_1\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}} \, _1F_1\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}} \, _1F_1\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 \, _1F_1\left (-\frac {b}{n a+a};\frac {n}{n+1};\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )} \\ y(x)\to \frac {b x^{n-1} \left (x^n\right )^{\frac {1}{n}} \, _1F_1\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 \, _1F_1\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*}