20.16.20 problem 20

Internal problem ID [3853]
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 : 20
Date solved : Monday, January 27, 2025 at 08:03:27 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-4 x_{1} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 27

dsolve([diff(x__1(t),t) = 4*x__2(t), diff(x__2(t),t) = -4*x__1(t), x__1(0) = 1, x__2(0) = 1], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \sin \left (4 t \right )+\cos \left (4 t \right ) \\ x_{2} \left (t \right ) &= \cos \left (4 t \right )-\sin \left (4 t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 28

DSolve[{D[x1[t],t]==4*x2[t],D[x2[t],t]==-4*x1[t]},{x1[0]==1,x2[0]==1},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \sin (4 t)+\cos (4 t) \\ \text {x2}(t)\to \cos (4 t)-\sin (4 t) \\ \end{align*}