14.30.3 problem 3

Internal problem ID [2816]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.3 (Stability of equilibrium solutions). Page 393
Problem number : 3
Date solved : Sunday, March 30, 2025 at 12:21:07 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.573 (sec). Leaf size: 257

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

\begin{align*} \\ \left [\left \{y \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{-2 t}}{4}, y \left (t \right ) = \frac {c_1 \,{\mathrm e}^{-2 t} \left (-\frac {8 c_2 \,c_1^{2} \left ({\mathrm e}^{-2 t}+2 c_2 \right )}{{\mathrm e}^{-4 t} c_1^{2}+4 \,{\mathrm e}^{-2 t} c_1^{2} c_2 +4 c_1^{2} c_2^{2}-64}-\frac {4 \,{\mathrm e}^{-2 t} c_1^{2} \left ({\mathrm e}^{-2 t}+2 c_2 \right )}{{\mathrm e}^{-4 t} c_1^{2}+4 \,{\mathrm e}^{-2 t} c_1^{2} c_2 +4 c_1^{2} c_2^{2}-64}+4\right )}{16}, y \left (t \right ) = -\frac {c_1 \,{\mathrm e}^{-2 t} \left (\frac {8 c_2 \,c_1^{2} \left ({\mathrm e}^{-2 t}+2 c_2 \right )}{{\mathrm e}^{-4 t} c_1^{2}+4 \,{\mathrm e}^{-2 t} c_1^{2} c_2 +4 c_1^{2} c_2^{2}-64}+\frac {4 \,{\mathrm e}^{-2 t} c_1^{2} \left ({\mathrm e}^{-2 t}+2 c_2 \right )}{{\mathrm e}^{-4 t} c_1^{2}+4 \,{\mathrm e}^{-2 t} c_1^{2} c_2 +4 c_1^{2} c_2^{2}-64}-4\right )}{16}\right \}, \left \{x \left (t \right ) = \frac {\frac {d}{d t}y \left (t \right )}{2 y \left (t \right )}\right \}\right ] \\ \end{align*}

✓ Mathematica. Time used: 0.254 (sec). Leaf size: 1239

ode={D[x[t],t]==x[t]^2+y[t]^2-1,D[y[t],t]==2*x[t]*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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