80.11.9 problem 9

Internal problem ID [21417]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 12. Stability theory. Excercise 12.6 at page 270
Problem number : 9
Date solved : Thursday, October 02, 2025 at 07:30:58 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-2 x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{2} \left (t \right )-2 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.072 (sec). Leaf size: 51

ode:=[diff(x__1(t),t) = -2*x__1(t)+x__2(t)+x__3(t), diff(x__2(t),t) = -2*x__2(t)+x__3(t), diff(x__3(t),t) = x__2(t)-2*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= 2 c_2 \,{\mathrm e}^{-t}+c_1 \,{\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-3 t} \\ x_{3} \left (t \right ) &= c_2 \,{\mathrm e}^{-t}-c_3 \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 91

ode={D[x1[t],t]==-2*x1[t]+x2[t]+x3[t],D[x2[t],t]==-2*x2[t]+x3[t],D[x3[t],t]==x2[t]-2*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to e^{-2 t} \left ((c_2+c_3) \left (e^t-1\right )+c_1\right )\\ \text {x2}(t)&\to \frac {1}{2} e^{-3 t} \left (c_2 \left (e^{2 t}+1\right )+c_3 \left (e^{2 t}-1\right )\right )\\ \text {x3}(t)&\to \frac {1}{2} e^{-3 t} \left (c_2 \left (e^{2 t}-1\right )+c_3 \left (e^{2 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.072 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
x3 = Function("x3") 
ode=[Eq(2*x1(t) - x2(t) - x3(t) + Derivative(x1(t), t),0),Eq(2*x2(t) - x3(t) + Derivative(x2(t), t),0),Eq(-x2(t) + 2*x3(t) + Derivative(x3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t),x3(t)],ics=ics)
 

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