61.19.11 problem 11

Internal problem ID [12201]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 11
Date solved : Wednesday, March 05, 2025 at 05:36:41 PM
CAS classification : [_Riccati]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

Timed Out