20.13.2 problem 2

Internal problem ID [3811]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 2
Date solved : Monday, January 27, 2025 at 08:02:53 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 30

dsolve([diff(x__1(t),t)=2*x__1(t)-3*x__2(t),diff(x__2(t),t)=x__1(t)-2*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 72

DSolve[{D[x1[t],t]==2*x1[t]-3*x2[t],D[x2[t],t]==x1[t]-2*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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