20.20.22 problem 22

Internal problem ID [3912]
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 : 22
Date solved : Monday, January 27, 2025 at 08:04:25 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 40

dsolve([diff(x__1(t),t)=-2*x__1(t)-0*x__2(t)-1*x__3(t),diff(x__2(t),t)=0*x__1(t)-1*x__2(t)-0*x__3(t),diff(x__3(t),t)=1*x__1(t)+0*x__2(t)-0*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 100

DSolve[{D[x1[t],t]==-2*x1[t]-0*x2[t]-1*x3[t],D[x2[t],t]==0*x1[t]-1*x2[t]-0*x3[t],D[x3[t],t]==1*x1[t]+0*x2[t]-0*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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