Internal problem ID [10379]
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]
\[ \boxed {y^{\prime } x -a \,x^{n} y^{2}-b y=c \,x^{-n}} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 69
dsolve(x*diff(y(x),x)=a*x^n*y(x)^2+b*y(x)+c*x^(-n),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {x^{-n} \left (b +n +\tan \left (\frac {\sqrt {4 a c -b^{2}-2 b n -n^{2}}\, \left (-\ln \left (x \right )+c_{1} \right )}{2}\right ) \sqrt {4 a c -b^{2}-2 b n -n^{2}}\right )}{2 a} \]
✓ Solution by Mathematica
Time used: 0.978 (sec). Leaf size: 138
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 (\frac {\sqrt {-4 a c+b^2+2 b n+n^2} \left (-x^{\sqrt {-4 a c+b^2+2 b n+n^2}}+c_1\right )}{x^{\sqrt {-4 a c+b^2+2 b n+n^2}}+c_1}-b-n\right )}{2 a} y(x)\to \frac {x^{-n} \left (\sqrt {-4 a c+b^2+2 b n+n^2}-b-n\right )}{2 a} \end{align*}