80.11.12 problem 12

Internal problem ID [21420]
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 : 12
Date solved : Thursday, October 02, 2025 at 07:31:05 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 162

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

\begin{align*} \text {x1}(t)&\to c_1 e^{a t}\\ \text {x2}(t)&\to \frac {1}{2} \left ((c_2-c_3) e^{(a-1) t}+(c_2+c_3) e^{(a+1) t}\right )\\ \text {x3}(t)&\to \frac {1}{2} \left ((c_3-c_2) e^{(a-1) t}+(c_2+c_3) e^{(a+1) t}\right )\\ \text {x1}(t)&\to 0\\ \text {x2}(t)&\to \frac {1}{2} \left ((c_2-c_3) e^{(a-1) t}+(c_2+c_3) e^{(a+1) t}\right )\\ \text {x3}(t)&\to \frac {1}{2} \left ((c_3-c_2) e^{(a-1) t}+(c_2+c_3) e^{(a+1) t}\right ) \end{align*}

✓ Sympy. Time used: 0.071 (sec). Leaf size: 46

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

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