80.11.10 problem 10

Internal problem ID [21418]
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 : 10
Date solved : Thursday, October 02, 2025 at 07:30:59 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=a x_{1} \left (t \right )+5 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-3 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 59

ode:=[diff(x__1(t),t) = a*x__1(t)+5*x__3(t), diff(x__2(t),t) = -x__2(t)-2*x__3(t), diff(x__3(t),t) = -3*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {\left (-a -3\right ) c_2 \,{\mathrm e}^{a t}}{a +3}-\frac {5 \,{\mathrm e}^{-3 t} c_3}{a +3} \\ x_{2} \left (t \right ) &= c_3 \,{\mathrm e}^{-3 t}+c_1 \,{\mathrm e}^{-t} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 75

ode={D[x1[t],t]==a*x1[t]+5*x3[t],D[x2[t],t]==-x2[t]-2*x3[t],D[x3[t],t]==3*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}-\frac {5 c_3 \left (e^{3 t}-e^{a t}\right )}{a-3}\\ \text {x2}(t)&\to \frac {1}{2} e^{-t} \left (-c_3 e^{4 t}+2 c_2+c_3\right )\\ \text {x3}(t)&\to c_3 e^{3 t} \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
a = symbols("a") 
x1 = Function("x1") 
x2 = Function("x2") 
x3 = Function("x3") 
ode=[Eq(-a*x1(t) - 5*x3(t) + Derivative(x1(t), t),0),Eq(x2(t) + 2*x3(t) + Derivative(x2(t), t),0),Eq(-3*x3(t) + Derivative(x3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t),x3(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = - \frac {5 C_{1} e^{3 t}}{a - 3} + C_{2} e^{a t}, \ x_{2}{\left (t \right )} = - \frac {C_{1} e^{3 t}}{2} + C_{3} e^{- t}, \ x_{3}{\left (t \right )} = C_{1} e^{3 t}\right ] \]