Internal problem ID [9626]
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: 39.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a \,x^{n} y^{2}-b y-c \,x^{-n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 90
dsolve(x*diff(y(x),x)=a*x^n*y(x)^2+b*y(x)+c*x^(-n),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\tan \left (-\frac {\ln \relax (x ) \sqrt {4 c a -b^{2}-2 b n -n^{2}}}{2}+\frac {c_{1} \sqrt {4 c a -b^{2}-2 b n -n^{2}}}{2}\right ) \sqrt {4 c a -b^{2}-2 b n -n^{2}}+b +n \right ) x^{-n}}{2 a} \]
✓ Solution by Mathematica
Time used: 0.583 (sec). Leaf size: 103
DSolve[x*y'[x]==a*x^n*y[x]^2+b*y[x]+c*x^(-n),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left (\sqrt {(b+n)^2-4 a c} \left (-1+\frac {2 c_1}{x^{\sqrt {(b+n)^2-4 a c}}+c_1}\right )-b-n\right )}{2 a} \\ y(x)\to \frac {x^{-n} \left (\sqrt {(b+n)^2-4 a c}-b-n\right )}{2 a} \\ \end{align*}