Internal problem ID [9590]
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]
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}-b \,x^{-n -2}=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 61
dsolve(diff(y(x),x)=a*x^n*y(x)^2+b*x^(-n-2),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {x^{-1-n} \left (n +1+\tan \left (\frac {\sqrt {4 b a -n^{2}-2 n -1}\, \left (-\ln \left (x \right )+c_{1} \right )}{2}\right ) \sqrt {4 b a -n^{2}-2 n -1}\right )}{2 a} \]
✓ Solution by Mathematica
Time used: 0.481 (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*}