64.25.2 problem 2

Internal problem ID [13626]
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 : 2
Date solved : Monday, March 31, 2025 at 08:03:04 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )+\frac {x \left (t \right ) \left (1-x \left (t \right )^{2}-y \left (t \right )^{2}\right )}{\sqrt {x \left (t \right )^{2}+y \left (t \right )^{2}}}\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+\frac {y \left (t \right ) \left (1-x \left (t \right )^{2}-y \left (t \right )^{2}\right )}{\sqrt {x \left (t \right )^{2}+y \left (t \right )^{2}}} \end{align*}

✗ Maple

ode:=[diff(x(t),t) = y(t)+x(t)/(x(t)^2+y(t)^2)^(1/2)*(1-x(t)^2-y(t)^2), diff(y(t),t) = -x(t)+y(t)/(x(t)^2+y(t)^2)^(1/2)*(1-x(t)^2-y(t)^2)]; 
dsolve(ode);
 

\[ \text {No solution found} \]

✗ Mathematica

ode={D[x[t],t]==y[t]+x[t]/Sqrt[x[t]^2+y[t]^2]*(1-(x[t]^2+y[t]^2)),D[y[t],t]==-x[t]+y[t]/Sqrt[x[t]^2+y[t]^2]*(1-(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(-y(t) + Derivative(x(t), t) - (-x(t)**2 - y(t)**2 + 1)*x(t)/sqrt(x(t)**2 + y(t)**2),0),Eq(x(t) + Derivative(y(t), t) - (-x(t)**2 - y(t)**2 + 1)*y(t)/sqrt(x(t)**2 + y(t)**2),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

Timed Out