20.16.13 problem 13

Internal problem ID [3846]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 13
Date solved : Monday, January 27, 2025 at 08:03:20 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 64

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 133

DSolve[{D[x1[t],t]==x1[t]+x2[t]-x3[t],D[x2[t],t]==x1[t]+x2[t]+x3[t],D[x3[t],t]==-x1[t]+x2[t]+x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (2 e^{3 t}+1\right )+(c_2-c_3) \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (2 e^{3 t}+1\right )+c_3 \left (e^{3 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{-t} \left (-\left (c_1 \left (e^{3 t}-1\right )\right )+c_2 \left (e^{3 t}-1\right )+c_3 \left (2 e^{3 t}+1\right )\right ) \\ \end{align*}