20.19.10 problem 11

Internal problem ID [3890]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.8 (Matrix exponential function), page 642
Problem number : 11
Date solved : Monday, January 27, 2025 at 08:04:06 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_{2} \left (t \right )-x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.336 (sec). Leaf size: 123

dsolve([diff(x__1(t),t)=0*x__1(t)-x__2(t)+0*x__3(t)+0*x__4(t),diff(x__2(t),t)=1*x__1(t)+0*x__2(t)-0*x__3(t)+0*x__4(t),diff(x__3(t),t)=1*x__2(t)+0*x__2(t)+0*x__3(t)-x__4(t),diff(x__4(t),t)=0*x__1(t)+1*x__2(t)+1*x__3(t)-0*x__4(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 ) &= c_4 \sin \left (t \right )-\cos \left (t \right ) c_3 \\ x_{3} \left (t \right ) &= \sin \left (t \right ) c_{2} +\cos \left (t \right ) c_{1} +\frac {\cos \left (t \right ) c_3}{2}+\frac {\sin \left (t \right ) c_3 t}{2}+\frac {c_4 \cos \left (t \right )}{2}+\frac {\sin \left (t \right ) c_4 t}{2}-\frac {\cos \left (t \right ) c_3 t}{2}+\frac {\cos \left (t \right ) c_4 t}{2} \\ x_{4} \left (t \right ) &= -c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )-\frac {\cos \left (t \right ) c_3 t}{2}-\frac {\cos \left (t \right ) c_4 t}{2}-\frac {\cos \left (t \right ) c_3}{2}-\frac {c_4 \cos \left (t \right )}{2}-\frac {\sin \left (t \right ) c_3 t}{2}+\frac {\sin \left (t \right ) c_4 t}{2}+c_4 \sin \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 80

DSolve[{D[x1[t],t]==0*x1[t]-1*x2[t]+0*x3[t]+0*x4[t],D[x2[t],t]==1*x1[t]+0*x2[t]-0*x3[t]+0*x4[t],D[x3[t],t]==1*x1[t]+0*x2[t]+0*x3[t]-x4[t],D[x4[t],t]==0*x1[t]+1*x2[t]+1*x3[t]-0*x4[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_1 t+c_3) \cos (t)-(c_2 t+c_4) \sin (t) \\ \text {x4}(t)\to (c_2 t+c_4) \cos (t)+(c_1 t+c_3) \sin (t) \\ \end{align*}