20.20.8 problem 8

Internal problem ID [3898]
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 : 8
Date solved : Monday, January 27, 2025 at 08:04:12 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )-7 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )+6 x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 60

dsolve([diff(x__1(t),t)=-3*x__1(t)-1*x__2(t)+0*x__3(t),diff(x__2(t),t)=4*x__1(t)-7*x__2(t)+0*x__3(t),diff(x__3(t),t)=6*x__1(t)+6*x__2(t)+4*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-5 t} \left (c_3 t +c_{2} \right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{-5 t} \left (2 c_3 t -c_3 +2 c_{2} \right ) \\ x_{3} \left (t \right ) &= \left (-\frac {2 \,{\mathrm e}^{-9 t} \left (9 c_3 t -2 c_3 +9 c_{2} \right )}{9}+c_{1} \right ) {\mathrm e}^{4 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 100

DSolve[{D[x1[t],t]==-3*x1[t]-1*x2[t]+0*x3[t],D[x2[t],t]==4*x1[t]-7*x2[t]+0*x3[t],D[x3[t],t]==6*x1[t]+6*x2[t]+4*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-5 t} (2 c_1 t-c_2 t+c_1) \\ \text {x2}(t)\to e^{-5 t} (4 c_1 t-2 c_2 t+c_2) \\ \text {x3}(t)\to \frac {1}{9} e^{-5 t} \left (2 c_1 \left (-18 t+5 e^{9 t}-5\right )+2 c_2 \left (9 t+2 e^{9 t}-2\right )+9 c_3 e^{9 t}\right ) \\ \end{align*}