28.4.33 problem 7.33

Internal problem ID [4565]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.33
Date solved : Sunday, March 30, 2025 at 03:26:23 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.191 (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) = x__1(t)+2*x__2(t)-x__3(t), diff(x__3(t),t) = x__1(t)-x__2(t)+2*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 99

ode={D[x1[t],t]==2*x1[t]-x2[t]+x3[t],D[x2[t],t]==x1[t]+2*x2[t]-x3[t],D[x3[t],t]==x1[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_1-(c_2-c_3) \left (e^t-1\right )\right ) \\ \text {x2}(t)\to e^t \left (c_1 \left (e^t-1\right )+(c_2-c_3) e^t+c_3\right ) \\ \text {x3}(t)\to e^t \left (c_1 \left (e^t-1\right )+(c_2-c_3) e^t+(c_3-c_2) e^{2 t}+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-2*x__1(t) + x__2(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - 2*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(-x__1(t) + x__2(t) - 2*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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