Internal problem ID [9627]
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: 40.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a \,x^{n} y^{2}-m y+a \,b^{2} x^{n +2 m}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
dsolve(x*diff(y(x),x)=a*x^n*y(x)^2+m*y(x)-a*b^2*x^(n+2*m),y(x), singsol=all)
\[ y \relax (x ) = i \tan \left (\frac {i x^{m +n} b a +c_{1} m +c_{1} n}{m +n}\right ) b \,x^{m} \]
✓ Solution by Mathematica
Time used: 2.667 (sec). Leaf size: 43
DSolve[x*y'[x]==a*x^n*y[x]^2+m*y[x]-a*b^2*x^(n+2*m),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {-b^2} x^m \tan \left (\frac {a \sqrt {-b^2} x^{m+n}}{m+n}+c_1\right ) \\ \end{align*}