20.16.5 problem 5

Internal problem ID [3838]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 5
Date solved : Monday, January 27, 2025 at 08:03:14 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 )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 51

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

\begin{align*} \text {x1}(t)\to e^{-t} (c_1 \cos (2 t)+c_2 \sin (2 t)) \\ \text {x2}(t)\to e^{-t} (c_2 \cos (2 t)-c_1 \sin (2 t)) \\ \end{align*}