Internal problem ID [10371]
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]
\[ \boxed {y^{\prime } x -x^{2 n} y^{2}-\left (m -n \right ) y=x^{2 m}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 34
dsolve(x*diff(y(x),x)=x^(2*n)*y(x)^2+(m-n)*y(x)+x^(2*m),y(x), singsol=all)
\[ y \left (x \right ) = \tan \left (\frac {x^{n +m}+\left (-n -m \right ) c_{1}}{n +m}\right ) x^{-n +m} \]
✓ Solution by Mathematica
Time used: 0.727 (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]
\[ y(x)\to x^{m-n} \tan \left (\frac {x^{m+n}}{m+n}+c_1\right ) \]