76.27.20 problem 20

Internal problem ID [17866]
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 : Tuesday, January 28, 2025 at 11:04:14 AM
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*}

✓ Solution by Maple

Time used: 0.126 (sec). Leaf size: 41

dsolve([diff(x__1(t),t)=0*x__1(t)+1*x__2(t)-1*x__3(t),diff(x__2(t),t)=1*x__1(t)+0*x__2(t)+1*x__3(t),diff(x__3(t),t)=1*x__1(t)+1*x__2(t)-0*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{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]},{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*}