20.19.6 problem 7

Internal problem ID [3886]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.8 (Matrix exponential function), page 642
Problem number : 7
Date solved : Monday, January 27, 2025 at 08:03:54 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{2} \left (t \right )-8 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{2} \left (t \right )-7 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 43

dsolve([diff(x__1(t),t)=2*x__1(t),diff(x__2(t),t)=x__2(t)-8*x__3(t),diff(x__3(t),t)=2*x__2(t)-7*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} c_3 \\ x_{2} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_{2} t +c_{1} \right ) \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (4 c_{2} t +4 c_{1} -c_{2} \right )}{8} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 106

DSolve[{D[x1[t],t]==2*x1[t],D[x2[t],t]==x2[t]-8*x3[t],D[x3[t],t]==2*x2[t]-7*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 e^{2 t} \\ \text {x2}(t)\to e^{-3 t} (4 c_2 t-8 c_3 t+c_2) \\ \text {x3}(t)\to e^{-3 t} (2 (c_2-2 c_3) t+c_3) \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to e^{-3 t} (4 c_2 t-8 c_3 t+c_2) \\ \text {x3}(t)\to e^{-3 t} (2 (c_2-2 c_3) t+c_3) \\ \end{align*}