76.27.20 problem 20

Internal problem ID [17795]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.5 (Fundamental Matrices and the Exponential of a Matrix). Problems at page 430
Problem number : 20
Date solved : Monday, March 31, 2025 at 04:27:52 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.160 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 94

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

\begin{align*} \text {x1}(t)\to e^{-t} \left ((c_1+c_2-c_3) e^t-c_2+c_3\right ) \\ \text {x2}(t)\to e^{-t} \left ((c_1+c_2) e^{2 t}-(c_1+c_2-c_3) e^t+c_2-c_3\right ) \\ \text {x3}(t)\to c_1 \left (e^t-1\right )+c_2 \left (e^t-1\right )+c_3 \\ \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 32

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

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