20.20.15 problem 15

Internal problem ID [3905]
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 : 15
Date solved : Monday, January 27, 2025 at 08:04:18 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-17 x_{1} \left (t \right )-42 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-7 x_{1} \left (t \right )+4 x_{2} \left (t \right )-14 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=7 x_{1} \left (t \right )+18 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 59

dsolve([diff(x__1(t),t)=-17*x__1(t)-0*x__2(t)-42*x__3(t),diff(x__2(t),t)=-7*x__1(t)+4*x__2(t)-14*x__3(t),diff(x__3(t),t)=7*x__1(t)+0*x__2(t)+18*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[{D[x1[t],t]==-17*x1[t]-0*x2[t]-42*x3[t],D[x2[t],t]==-7*x1[t]+4*x2[t]-14*x3[t],D[x3[t],t]==7*x1[t]+0*x2[t]+18*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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