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
:
Sunday, March 30, 2025 at 12:20:55 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: 0.073 (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: 0.138 (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)
\[
\left [ x{\left (t \right )} = C_{1} t \sin {\left (2 t \right )} + C_{2} t \cos {\left (2 t \right )} + C_{3} \sin {\left (2 t \right )} + C_{4} \cos {\left (2 t \right )}, \ y{\left (t \right )} = C_{1} t \cos {\left (2 t \right )} - C_{2} t \sin {\left (2 t \right )} + C_{3} \cos {\left (2 t \right )} - C_{4} \sin {\left (2 t \right )}, \ z{\left (t \right )} = C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )}, \ h{\left (t \right )} = C_{1} \cos {\left (2 t \right )} - C_{2} \sin {\left (2 t \right )}\right ]
\]