14.29.10 problem 10

Internal problem ID [2808]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 10
Date solved : Monday, January 27, 2025 at 06:14:05 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 y+z \left (t \right )\\ y^{\prime }&=-2 x \left (t \right )+h \left (t \right )\\ z^{\prime }\left (t \right )&=2 h \left (t \right )\\ h^{\prime }\left (t \right )&=-2 z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.245 (sec). Leaf size: 110

dsolve([diff(x(t),t)=2*y(t)+z(t),diff(y(t),t)=-2*x(t)+h(t),diff(z(t),t)=2*h(t),diff(h(t),t)=-2*z(t)],singsol=all)
 

\begin{align*} h \left (t \right ) &= c_3 \sin \left (2 t \right )+c_4 \cos \left (2 t \right ) \\ x \left (t \right ) &= c_2 \sin \left (2 t \right )+c_1 \cos \left (2 t \right )+\frac {c_4 \cos \left (2 t \right )}{2}+\sin \left (2 t \right ) c_4 t -\cos \left (2 t \right ) c_3 t \\ y &= c_2 \cos \left (2 t \right )-c_1 \sin \left (2 t \right )+\cos \left (2 t \right ) c_4 t +\sin \left (2 t \right ) c_3 t -\frac {c_4 \sin \left (2 t \right )}{2} \\ z \left (t \right ) &= -c_3 \cos \left (2 t \right )+c_4 \sin \left (2 t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.073 (sec). Leaf size: 96

DSolve[{D[x[t],t]==2*y[t]+z[t],D[y[t],t]==-2*x[t]+h[t],D[z[t],t]==2*h[t],D[h[t],t]==-2*z[t]},{x[t],y[t],z[t],h[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} h(t)\to c_1 \cos (2 t)-c_4 \sin (2 t) \\ x(t)\to (c_4 t+c_2) \cos (2 t)+(c_1 t+c_3) \sin (2 t) \\ y(t)\to (c_1 t+c_3) \cos (2 t)-(c_4 t+c_2) \sin (2 t) \\ z(t)\to c_4 \cos (2 t)+c_1 \sin (2 t) \\ \end{align*}