20.19.3 problem 4

Internal problem ID [3883]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.8 (Matrix exponential function), page 642
Problem number : 4
Date solved : Tuesday, March 04, 2025 at 05:18:37 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 102

ode={D[x1[t],t]==3*x1[t],D[x2[t],t]==3*x2[t]-x3[t],D[x3[t],t]==x2[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^{3 t} \\ \text {x2}(t)\to e^{2 t} (c_2 (t+1)-c_3 t) \\ \text {x3}(t)\to e^{2 t} ((c_2-c_3) t+c_3) \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to e^{2 t} (c_2 (t+1)-c_3 t) \\ \text {x3}(t)\to e^{2 t} ((c_2-c_3) t+c_3) \\ \end{align*}

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

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*x__1(t) + Derivative(x__1(t), t),0),Eq(-3*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(-x__2(t) - x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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