8.29.10 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 : Tuesday, September 30, 2025 at 05:52:52 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.402 (sec). Leaf size: 110

ode:=[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)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 96

ode={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]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 95

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
h = Function("h") 
ode=[Eq(-2*y(t) - z(t) + Derivative(x(t), t),0),Eq(-h(t) + 2*x(t) + Derivative(y(t), t),0),Eq(-2*h(t) + Derivative(z(t), t),0),Eq(2*z(t) + Derivative(h(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t),h(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} t \sin {\left (2 t \right )} + C_{2} t \cos {\left (2 t \right )} + C_{3} \sin {\left (2 t \right )} + C_{4} \cos {\left (2 t \right )}, \ y{\left (t \right )} = C_{1} t \cos {\left (2 t \right )} - C_{2} t \sin {\left (2 t \right )} + C_{3} \cos {\left (2 t \right )} - C_{4} \sin {\left (2 t \right )}, \ z{\left (t \right )} = C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )}, \ h{\left (t \right )} = C_{1} \cos {\left (2 t \right )} - C_{2} \sin {\left (2 t \right )}\right ] \]