64.25.2 problem 2

Internal problem ID [13697]
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 : Tuesday, January 28, 2025 at 05:55:24 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*}

✗ Solution by Maple

dsolve([diff(x(t),t)=y(t)+x(t)/sqrt(x(t)^2+y(t)^2)*(1-(x(t)^2+y(t)^2)),diff(y(t),t)=-x(t)+y(t)/sqrt(x(t)^2+y(t)^2)*(1-(x(t)^2+y(t)^2))],singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[{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))},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

Not solved