61.2.29 problem 29

Internal problem ID [11956]
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 : 29
Date solved : Sunday, March 30, 2025 at 09:28:31 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=-\left (n +1\right ) x^{n} y^{2}+a \,x^{n +m +1}-a \,x^{m} \end{align*}

✗ Maple

ode:=diff(y(x),x) = -(n+1)*x^n*y(x)^2+a*x^(m+n+1)-a*x^m; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[y[x],x]==-(n+1)*x^n*y[x]^2+a*x^(n+m+1)-a*x^m; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a*x**m - a*x**(m + n + 1) + x**n*(n + 1)*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a*x**m - a*x**(m + n + 1) + n*x**n*y(x)**2 + x**n*y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method