64.25.1 problem 1

Internal problem ID [13625]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 13, Limit cycles and periodic solutions. Section 13.4, Exercises page 706
Problem number : 1
Date solved : Monday, March 31, 2025 at 08:03:03 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )-4 y \left (t \right )-x \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )+4 y \left (t \right )-y \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \end{align*}

✗ Maple

ode:=[diff(x(t),t) = 4*x(t)-4*y(t)-x(t)*(x(t)^2+y(t)^2), diff(y(t),t) = 4*x(t)+4*y(t)-y(t)*(x(t)^2+y(t)^2)]; 
dsolve(ode);
 

\[ \text {No solution found} \]

✗ Mathematica

ode={D[x[t],t]==4*x[t]-4*y[t]-x[t]*(x[t]^2+y[t]^2),D[y[t],t]==4*x[t]+4*y[t]-y[t]*(x[t]^2+y[t]^2)}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq((x(t)**2 + y(t)**2)*x(t) - 4*x(t) + 4*y(t) + Derivative(x(t), t),0),Eq((x(t)**2 + y(t)**2)*y(t) - 4*x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

Timed Out