Internal problem ID [9631]
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: 44.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a \,x^{m +2 n} y^{2}-\left (x^{m +n} b -n \right ) y-c \,x^{m}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 87
dsolve(x*diff(y(x),x)=a*x^(2*n+m)*y(x)^2+(b*x^(n+m)-n)*y(x)+c*x^m,y(x), singsol=all)
\[ y \relax (x ) = \frac {x^{m -1} \left (\sqrt {4 a \,b^{2} c -b^{4}}\, \tan \left (\frac {\sqrt {4 a \,b^{2} c -b^{4}}\, \left (b \,x^{m +n}+c_{1} m +c_{1} n \right )}{2 b^{2} \left (m +n \right )}\right )-b^{2}\right ) x^{-m -n +1}}{2 a b} \]
✓ Solution by Mathematica
Time used: 0.952 (sec). Leaf size: 102
DSolve[x*y'[x]==a*x^(2*n+m)*y[x]^2+(b*x^(n+m)-n)*y[x]+c*x^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-n} \left (-b+\sqrt {b^2-4 a c} \left (-1+\frac {2 c_1}{e^{\frac {\sqrt {b^2-4 a c} x^{m+n}}{m+n}}+c_1}\right )\right )}{2 a} \\ y(x)\to \frac {x^{-n} \left (\sqrt {b^2-4 a c}-b\right )}{2 a} \\ \end{align*}