8.29.8 problem 8

Internal problem ID [2806]
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 : 8
Date solved : Tuesday, September 30, 2025 at 05:52:51 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 x \left (t \right )+y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-3 x \left (t \right )+2 y \left (t \right )+3 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )-y \left (t \right )-2 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.169 (sec). Leaf size: 53

ode:=[diff(x(t),t) = -2*x(t)+y(t)+z(t), diff(y(t),t) = -3*x(t)+2*y(t)+3*z(t), diff(z(t),t) = x(t)-y(t)-2*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_2 +c_3 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 3 c_2 +3 c_3 \,{\mathrm e}^{-t}+{\mathrm e}^{-t} c_1 \\ z \left (t \right ) &= -2 c_3 \,{\mathrm e}^{-t}-c_2 -{\mathrm e}^{-t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 106

ode={D[x[t],t]==-2*x[t]+1*y[t]+1*z[t],D[y[t],t]==-3*x[t]+2*y[t]+3*z[t],D[z[t],t]==1*x[t]-1*y[t]-2*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t} \left ((c_2+c_3) \left (e^t-1\right )-c_1 \left (e^t-2\right )\right )\\ y(t)&\to e^{-t} \left (-3 c_1 \left (e^t-1\right )+c_2 \left (3 e^t-2\right )+3 c_3 \left (e^t-1\right )\right )\\ z(t)&\to e^{-t} \left (c_1 \left (e^t-1\right )-c_2 \left (e^t-1\right )-c_3 \left (e^t-2\right )\right ) \end{align*}

✓ Sympy. Time used: 0.060 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(2*x(t) - y(t) - z(t) + Derivative(x(t), t),0),Eq(3*x(t) - 2*y(t) - 3*z(t) + Derivative(y(t), t),0),Eq(-x(t) + y(t) + 2*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} + \left (C_{2} + C_{3}\right ) e^{- t}, \ y{\left (t \right )} = - 3 C_{1} + C_{2} e^{- t}, \ z{\left (t \right )} = C_{1} + C_{3} e^{- t}\right ] \]