87.19.11 problem 15
Internal
problem
ID
[23681]
Book
:
Ordinary
differential
equations
with
modern
applications.
Ladas,
G.
E.
and
Finizio,
N.
Wadsworth
Publishing.
California.
1978.
ISBN
0-534-00552-7.
QA372.F56
Section
:
Chapter
3.
Linear
Systems.
Exercise
at
page
149
Problem
number
:
15
Date
solved
:
Thursday, October 02, 2025 at 09:44:12 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&=y \left (t \right )-z \left (t \right )\\ y^{\prime }\left (t \right )&=-x+z \left (t \right )\\ z^{\prime }\left (t \right )&=x-y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.062 (sec). Leaf size: 116
ode:=[diff(x(t),t) = y(t)-z(t), diff(y(t),t) = z(t)-x(t), diff(z(t),t) = x(t)-y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 +c_2 \sin \left (\sqrt {3}\, t \right )+c_3 \cos \left (\sqrt {3}\, t \right ) \\
y \left (t \right ) &= -\frac {c_2 \sin \left (\sqrt {3}\, t \right )}{2}-\frac {c_3 \cos \left (\sqrt {3}\, t \right )}{2}+c_1 +\frac {c_2 \sqrt {3}\, \cos \left (\sqrt {3}\, t \right )}{2}-\frac {c_3 \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )}{2} \\
z \left (t \right ) &= -\frac {c_2 \sqrt {3}\, \cos \left (\sqrt {3}\, t \right )}{2}+\frac {c_3 \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )}{2}-\frac {c_2 \sin \left (\sqrt {3}\, t \right )}{2}-\frac {c_3 \cos \left (\sqrt {3}\, t \right )}{2}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 0.02 (sec). Leaf size: 168
ode={D[x[t],t]==y[t]-z[t],D[y[t],t]==z[t]-x[t],D[z[t],t]==x[t]-y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{3} \left ((2 c_1-c_2-c_3) \cos \left (\sqrt {3} t\right )+\sqrt {3} (c_2-c_3) \sin \left (\sqrt {3} t\right )+c_1+c_2+c_3\right )\\ y(t)&\to \frac {1}{3} \left (-(c_1-2 c_2+c_3) \cos \left (\sqrt {3} t\right )-\sqrt {3} (c_1-c_3) \sin \left (\sqrt {3} t\right )+c_1+c_2+c_3\right )\\ z(t)&\to \frac {1}{3} \left (-(c_1+c_2-2 c_3) \cos \left (\sqrt {3} t\right )+\sqrt {3} (c_1-c_2) \sin \left (\sqrt {3} t\right )+c_1+c_2+c_3\right ) \end{align*}
✓ Sympy. Time used: 0.165 (sec). Leaf size: 112
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-y(t) + z(t) + Derivative(x(t), t),0),Eq(x(t) - z(t) + Derivative(y(t), t),0),Eq(-x(t) + y(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{1} - \left (\frac {C_{2}}{2} + \frac {\sqrt {3} C_{3}}{2}\right ) \cos {\left (\sqrt {3} t \right )} - \left (\frac {\sqrt {3} C_{2}}{2} - \frac {C_{3}}{2}\right ) \sin {\left (\sqrt {3} t \right )}, \ y{\left (t \right )} = C_{1} - \left (\frac {C_{2}}{2} - \frac {\sqrt {3} C_{3}}{2}\right ) \cos {\left (\sqrt {3} t \right )} + \left (\frac {\sqrt {3} C_{2}}{2} + \frac {C_{3}}{2}\right ) \sin {\left (\sqrt {3} t \right )}, \ z{\left (t \right )} = C_{1} + C_{2} \cos {\left (\sqrt {3} t \right )} - C_{3} \sin {\left (\sqrt {3} t \right )}\right ]
\]