Internal problem ID [9594]
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: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \,x^{n} y^{2}-b \,x^{-n -2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 77
dsolve(diff(y(x),x)=a*x^n*y(x)^2+b*x^(-n-2),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\tan \left (-\frac {\ln \relax (x ) \sqrt {4 b a -n^{2}-2 n -1}}{2}+\frac {c_{1} \sqrt {4 b a -n^{2}-2 n -1}}{2}\right ) \sqrt {4 b a -n^{2}-2 n -1}+n +1\right ) x^{-n -1}}{2 a} \]
✓ Solution by Mathematica
Time used: 0.431 (sec). Leaf size: 135
DSolve[y'[x]==a*x^n*y[x]^2+b*x^(-n-2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n-1} \left (-\left (\sqrt {(n+1)^2-4 a b}+n+1\right ) x^{\sqrt {(n+1)^2-4 a b}}+c_1 \left (\sqrt {(n+1)^2-4 a b}-n-1\right )\right )}{2 a \left (x^{\sqrt {(n+1)^2-4 a b}}+c_1\right )} \\ y(x)\to \frac {x^{-n-1} \left (\sqrt {(n+1)^2-4 a b}-n-1\right )}{2 a} \\ \end{align*}