14.22.9 problem 9

Internal problem ID [2736]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page 339
Problem number : 9
Date solved : Tuesday, March 04, 2025 at 02:40:34 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = -2\\ x_{3} \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 26

ode:=[diff(x__1(t),t) = 3*x__1(t)+x__2(t)-x__3(t), diff(x__2(t),t) = x__1(t)+3*x__2(t)-x__3(t), diff(x__3(t),t) = 3*x__1(t)+3*x__2(t)-x__3(t)]; 
ic:=x__1(0) = 1x__2(0) = -2x__3(0) = -1; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 30

ode={D[ x1[t],t]==3*x1[t]+1*x2[t]-1*x3[t],D[ x2[t],t]==1*x1[t]+3*x2[t]-1*x3[t],D[ x3[t],t]==3*x1[t]+3*x2[t]-1*x3[t]}; 
ic={x1[0]==1,x2[0]==-2,x3[0]==-1}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{2 t} \\ \text {x2}(t)\to -2 e^{2 t} \\ \text {x3}(t)\to -e^{2 t} \\ \end{align*}

✓ Sympy. Time used: 0.128 (sec). Leaf size: 46

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) - x__2(t) + x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - 3*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(-3*x__1(t) - 3*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)
 

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