Internal problem ID [9628]
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: 41.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -x^{2 n} y^{2}-\left (-n +m \right ) y-x^{2 m}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 35
dsolve(x*diff(y(x),x)=x^(2*n)*y(x)^2+(m-n)*y(x)+x^(2*m),y(x), singsol=all)
\[ y \relax (x ) = \tan \left (\frac {-c_{1} m -c_{1} n +x^{n +m}}{n +m}\right ) x^{m -n} \]
✓ Solution by Mathematica
Time used: 0.842 (sec). Leaf size: 28
DSolve[x*y'[x]==x^(2*n)*y[x]^2+(m-n)*y[x]+x^(2*m),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^{m-n} \tan \left (\frac {x^{m+n}}{m+n}+c_1\right ) \\ \end{align*}