80.11.23 problem 24

Internal problem ID [21431]
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 : 24
Date solved : Thursday, October 02, 2025 at 07:31:30 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 177

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

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

✓ Sympy. Time used: 0.040 (sec). Leaf size: 22

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

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