14.29.9 problem 9

Internal problem ID [2807]
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 : 9
Date solved : Monday, January 27, 2025 at 06:14:04 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 y\\ y^{\prime }&=-2 x \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: 0.034 (sec). Leaf size: 67

dsolve([diff(x(t),t)=0*x(t)+2*y(t),diff(y(t),t)=-2*x(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_1 \sin \left (2 t \right )+c_2 \cos \left (2 t \right ) \\ y &= c_1 \cos \left (2 t \right )-c_2 \sin \left (2 t \right ) \\ 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: 0.003 (sec). Leaf size: 76

DSolve[{D[x[t],t]==0*x[t]+2*y[t],D[y[t],t]==-2*x[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*} x(t)\to c_1 \cos (2 t)+c_2 \sin (2 t) \\ y(t)\to c_2 \cos (2 t)-c_1 \sin (2 t) \\ h(t)\to c_3 \cos (2 t)-c_4 \sin (2 t) \\ z(t)\to c_4 \cos (2 t)+c_3 \sin (2 t) \\ \end{align*}