Internal problem ID [10355]
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]
\[ \boxed {y^{\prime }-y^{2}-a \,x^{n} y=a \,x^{n -1}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 400
dsolve(diff(y(x),x)=y(x)^2+a*x^n*y(x)+a*x^(n-1),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-{\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (-\frac {a x \,x^{n}}{n +1}\right )^{-\frac {n}{2 n +2}} \left (n +1\right )^{2} \left (x^{n} a x -n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )-2 n \left (-\frac {\left (n +1\right ) n \,{\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (-\frac {a x \,x^{n}}{n +1}\right )^{-\frac {n}{2 n +2}} \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )}{2}+\left (n +\frac {1}{2}\right ) x \,x^{n} a \left (c_{1} x -{\mathrm e}^{\frac {a x \,x^{n}}{n +1}}\right )\right )}{x \left ({\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (-\frac {a x \,x^{n}}{n +1}\right )^{-\frac {n}{2 n +2}} \left (n +1\right )^{2} \left (x^{n} a x -n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )+2 n \left (-\frac {\left (n +1\right ) n \,{\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (-\frac {a x \,x^{n}}{n +1}\right )^{-\frac {n}{2 n +2}} \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )}{2}+a \,x^{2} c_{1} x^{n} \left (n +\frac {1}{2}\right )\right )\right )} \]
✓ Solution by Mathematica
Time used: 2.82 (sec). Leaf size: 136
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 {\left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}} \Gamma \left (-\frac {1}{n+1},-\frac {a x^{n+1}}{n+1}\right )-(n+1) \left (e^{\frac {a x^{n+1}}{n+1}}+c_1 x\right )}{x \left (-\left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}} \Gamma \left (-\frac {1}{n+1},-\frac {a x^{n+1}}{n+1}\right )+c_1 (n+1) x\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}