20.19.2 problem 3

Internal problem ID [3882]
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 : 3
Date solved : Monday, January 27, 2025 at 08:03:51 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 25

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 35

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

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