20.16.16 problem 16

Internal problem ID [3849]
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 : 16
Date solved : Monday, January 27, 2025 at 08:03:23 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 51

dsolve([diff(x__1(t),t)=x__2(t),diff(x__2(t),t)=-x__1(t),diff(x__3(t),t)=-x__4(t),diff(x__4(t),t)=x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_3 \sin \left (t \right )+c_4 \cos \left (t \right ) \\ x_{2} \left (t \right ) &= \cos \left (t \right ) c_3 -c_4 \sin \left (t \right ) \\ x_{3} \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ x_{4} \left (t \right ) &= \sin \left (t \right ) c_{2} -\cos \left (t \right ) c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 60

DSolve[{D[x1[t],t]==x2[t],D[x2[t],t]==-x1[t],D[x3[t],t]==-x4[t],D[x4[t],t]==x3[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (t)+c_2 \sin (t) \\ \text {x2}(t)\to c_2 \cos (t)-c_1 \sin (t) \\ \text {x3}(t)\to c_3 \cos (t)-c_4 \sin (t) \\ \text {x4}(t)\to c_4 \cos (t)+c_3 \sin (t) \\ \end{align*}