61.2.39 problem 39

Internal problem ID [12045]
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
Date solved : Monday, January 27, 2025 at 11:53:24 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Riccati]

\begin{align*} x y^{\prime }&=a \,x^{n} y^{2}+b y+c \,x^{-n} \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 73

dsolve(x*diff(y(x),x)=a*x^n*y(x)^2+b*y(x)+c*x^(-n),y(x), singsol=all)
 

\[ y = \frac {x^{-n} \left (\tan \left (\frac {\sqrt {4 a c -b^{2}-2 b n -n^{2}}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {4 a c -b^{2}-2 b n -n^{2}}-b -n \right )}{2 a} \]

✓ Solution by Mathematica

Time used: 0.575 (sec). Leaf size: 138

DSolve[x*D[y[x],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*}