8.29.7 problem 7

Internal problem ID [2805]
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 : 7
Date solved : Tuesday, September 30, 2025 at 05:52:50 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.123 (sec). Leaf size: 72

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 87

ode={D[x[t],t]==0*x[t]+2*y[t]+1*z[t],D[y[t],t]==-1*x[t]-3*y[t]-1*z[t],D[z[t],t]==1*x[t]+1*y[t]-1*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.086 (sec). Leaf size: 56

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

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