20.19.8 problem 9

Internal problem ID [3888]
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 : 9
Date solved : Monday, January 27, 2025 at 08:04:05 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-8 x_{1} \left (t \right )+6 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-12 x_{1} \left (t \right )+10 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.079 (sec). Leaf size: 73

dsolve([diff(x__1(t),t)=-8*x__1(t)+6*x__2(t)-3*x__3(t),diff(x__2(t),t)=-12*x__1(t)+10*x__2(t)-3*x__3(t),diff(x__3(t),t)=-12*x__2(t)+12*x__2(t)-2*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 105

DSolve[{D[x1[t],t]==-8*x1[t]+6*x2[t]-3*x3[t],D[x2[t],t]==-12*x1[t]+10*x2[t]-3*x3[t],D[x3[t],t]==-12*x1[t]+12*x2[t]-2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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