20.16.3 problem 3

Internal problem ID [3836]
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 : 3
Date solved : Monday, January 27, 2025 at 08:03:12 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*}

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 39

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

\begin{align*} \text {x1}(t)\to c_1 \cos (4 t)-c_2 \sin (4 t) \\ \text {x2}(t)\to c_2 \cos (4 t)+c_1 \sin (4 t) \\ \end{align*}