Internal problem ID [10335]
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: 5.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}=a n \,x^{n -1}-a^{2} x^{2 n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 388
dsolve(diff(y(x),x)=y(x)^2+a*n*x^(n-1)-a^2*x^(2*n),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-3 \left (n +2\right ) c_{1} \left (\left (\frac {1}{3} n^{2}+n +\frac {2}{3}\right ) x^{-\frac {3 n}{2}}+a x \,x^{-\frac {n}{2}} \left (n +\frac {4}{3}\right )\right ) {\mathrm e}^{\frac {a x \,x^{n}}{n +1}} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x \,x^{n}}{n +1}\right )+2 c_{1} {\mathrm e}^{\frac {a x \,x^{n}}{n +1}} \left (n +1\right ) \left (\left (-\frac {1}{2} n^{2}-\frac {3}{2} n -1\right ) x^{-\frac {3 n}{2}}+x a \left (\left (-\frac {n}{2}-1\right ) x^{-\frac {n}{2}}+a x \,x^{\frac {n}{2}}\right )\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x \,x^{n}}{n +1}\right )+2 \left (n +2\right )^{2} c_{1} \left (n +\frac {3}{2}\right ) {\mathrm e}^{\frac {2 a x \,x^{n}}{n +1}} x^{-\frac {3 n}{2}} \left (-\frac {2 a x \,x^{n}}{n +1}\right )^{\frac {3 n +4}{2 n +2}}+2 x^{2} a \,x^{n}}{2 \left (-\frac {{\mathrm e}^{\frac {a x \,x^{n}}{n +1}} x^{-\frac {3 n}{2}} c_{1} \left (n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x \,x^{n}}{n +1}\right )}{2}+c_{1} \left (\left (-\frac {n}{2}-1\right ) x^{-\frac {3 n}{2}}+a x \,x^{-\frac {n}{2}}\right ) {\mathrm e}^{\frac {a x \,x^{n}}{n +1}} \left (n +1\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x \,x^{n}}{n +1}\right )+x \right ) x} \]
✓ Solution by Mathematica
Time used: 1.61 (sec). Leaf size: 227
DSolve[y'[x]==y[x]^2+a*n*x^(n-1)-a^2*x^(2*n),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2^{\frac {1}{n+1}} (n+1) \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}} \left (a x^n-c_1 e^{\frac {2 a x^{n+1}}{n+1}}\right )-a c_1 x^{n+1} \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}{2^{\frac {1}{n+1}} (n+1) \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}}-c_1 x \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )} \\ y(x)\to \frac {2^{\frac {1}{n+1}} (n+1) e^{\frac {2 a x^{n+1}}{n+1}} \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}}}{x \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}+a x^n \\ \end{align*}