14.29.1 problem 1

Internal problem ID [2799]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 1
Date solved : Tuesday, March 04, 2025 at 02:42:47 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 26

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

\begin{align*} x \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{-t} \\ y &= -2 c_2 \,{\mathrm e}^{-t}-c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 58

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

\begin{align*} x(t)\to e^{-t} \left (c_1 \left (2 e^t-1\right )+c_2 \left (e^t-1\right )\right ) \\ y(t)\to e^{-t} \left (-2 c_1 \left (e^t-1\right )-c_2 \left (e^t-2\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.073 (sec). Leaf size: 20

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

\[ \left [ x{\left (t \right )} = - C_{1} - \frac {C_{2} e^{- t}}{2}, \ y{\left (t \right )} = C_{1} + C_{2} e^{- t}\right ] \]