Internal problem ID [9612]
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: 25.
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-x^{-1+n} a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 385
dsolve(diff(y(x),x)=y(x)^2+a*x^n*y(x)+a*x^(n-1),y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\frac {x^{n +1} a}{n +1}}}{x^{2} \left (c_{1}-\frac {\left (\frac {a}{-n -1}\right )^{\frac {1}{n +1}} \left (\frac {\left (-n -1\right )^{2} x^{-\frac {n}{n +1}-\frac {1}{n +1}-n -1} \left (\frac {a}{-n -1}\right )^{-\frac {1}{n +1}} \left (\frac {x^{n +1} a \,n^{2}}{-n -1}+\frac {2 x^{n +1} a n}{-n -1}+n^{2}+\frac {x^{n +1} a}{-n -1}+n \right ) \left (\frac {x^{n +1} a}{-n -1}\right )^{-\frac {n}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {x^{n +1} a}{2 \left (-n -1\right )}} \WhittakerM \left (-\frac {1}{n +1}-\frac {n}{2 \left (n +1\right )}, \frac {n}{2 n +2}+\frac {1}{2}, \frac {x^{n +1} a}{-n -1}\right )}{n \left (2 n +1\right ) a}+\frac {\left (-n -1\right )^{2} x^{-\frac {n}{n +1}-\frac {1}{n +1}-n -1} \left (\frac {a}{-n -1}\right )^{-\frac {1}{n +1}} n \left (\frac {x^{n +1} a}{-n -1}\right )^{-\frac {n}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {x^{n +1} a}{2 \left (-n -1\right )}} \WhittakerM \left (-\frac {1}{n +1}-\frac {n}{2 \left (n +1\right )}+1, \frac {n}{2 n +2}+\frac {1}{2}, \frac {x^{n +1} a}{-n -1}\right )}{\left (2 n +1\right ) a}\right )}{n +1}\right )}-\frac {1}{x} \]
✓ Solution by Mathematica
Time used: 1.653 (sec). Leaf size: 72
DSolve[y'[x]==y[x]^2+a*x^n*y[x]+a*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1+\frac {(n+1) e^{\frac {a x^{n+1}}{n+1}}}{-E_{1+\frac {1}{n+1}}\left (-\frac {a x^{n+1}}{n+1}\right )+c_1 (n+1) x}}{x} \\ y(x)\to -\frac {1}{x} \\ \end{align*}