20.15.3 problem 3

Internal problem ID [3829]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.3, page 598
Problem number : 3
Date solved : Monday, January 27, 2025 at 08:03:07 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: 0.019 (sec). Leaf size: 41

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: 47

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*}