20.20.13 problem 13

Internal problem ID [3903]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 13
Date solved : Monday, January 27, 2025 at 08:04:16 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-2 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )+2 x_{2} \left (t \right )-3 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 66

dsolve([diff(x__1(t),t)=2*x__1(t)-2*x__2(t)+1*x__3(t),diff(x__2(t),t)=1*x__1(t)-4*x__2(t)+1*x__3(t),diff(x__3(t),t)=2*x__1(t)+2*x__2(t)-3*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-5 t}+{\mathrm e}^{2 t} c_3 \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t} c_3}{4}+c_{1} {\mathrm e}^{-2 t}+2 c_{2} {\mathrm e}^{-5 t} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{2 t} c_3}{2}+2 c_{1} {\mathrm e}^{-2 t}-3 c_{2} {\mathrm e}^{-5 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 153

DSolve[{D[x1[t],t]==2*x1[t]-2*x2[t]+1*x3[t],D[x2[t],t]==1*x1[t]-4*x2[t]+1*x3[t],D[x3[t],t]==2*x1[t]+2*x2[t]-3*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{7} e^{-5 t} \left ((7 c_1-2 c_2+c_3) e^{7 t}+2 c_2-c_3\right ) \\ \text {x2}(t)\to \frac {1}{28} e^{-5 t} \left (-7 (c_1-2 c_2-c_3) e^{3 t}+(7 c_1-2 c_2+c_3) e^{7 t}+16 c_2-8 c_3\right ) \\ \text {x3}(t)\to \frac {1}{14} e^{-5 t} \left (-7 (c_1-2 c_2-c_3) e^{3 t}+(7 c_1-2 c_2+c_3) e^{7 t}+6 (c_3-2 c_2)\right ) \\ \end{align*}