60.3.3 problem Problem 4
Internal
problem
ID
[15227]
Book
:
Differential
equations
and
the
calculus
of
variations
by
L.
ElSGOLTS.
MIR
PUBLISHERS,
MOSCOW,
Third
printing
1977.
Section
:
Chapter
3,
SYSTEMS
OF
DIFFERENTIAL
EQUATIONS.
Problems
page
209
Problem
number
:
Problem
4
Date
solved
:
Thursday, October 02, 2025 at 10:07:33 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right ) \end{align*}
✓ Maple. Time used: 0.146 (sec). Leaf size: 175
ode:=[diff(x(t),t) = y(t), diff(y(t),t) = z(t), diff(z(t),t) = x(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right )+c_3 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) \\
y \left (t \right ) &= c_1 \,{\mathrm e}^{t}-\frac {c_2 \,{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{2}+\frac {c_2 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{2} \\
z \left (t \right ) &= c_1 \,{\mathrm e}^{t}-\frac {c_2 \,{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{2}-\frac {c_2 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )}{2}+\frac {c_3 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.018 (sec). Leaf size: 234
ode={D[x[t],t]==y[t],D[y[t],t]==z[t],D[z[t],t]==x[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{3} e^{-t/2} \left ((c_1+c_2+c_3) e^{3 t/2}+(2 c_1-c_2-c_3) \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (c_2-c_3) \sin \left (\frac {\sqrt {3} t}{2}\right )\right )\\ y(t)&\to \frac {1}{3} e^{-t/2} \left ((c_1+c_2+c_3) e^{3 t/2}-(c_1-2 c_2+c_3) \cos \left (\frac {\sqrt {3} t}{2}\right )-\sqrt {3} (c_1-c_3) \sin \left (\frac {\sqrt {3} t}{2}\right )\right )\\ z(t)&\to \frac {1}{3} e^{-t/2} \left ((c_1+c_2+c_3) e^{3 t/2}-(c_1+c_2-2 c_3) \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (c_1-c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}
✓ Sympy. Time used: 0.169 (sec). Leaf size: 163
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
z = Function("z")
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(-z(t) + Derivative(y(t), t),0),Eq(-x(t) + Derivative(z(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
\[
\left [ x{\left (t \right )} = C_{3} e^{t} - \left (\frac {C_{1}}{2} + \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - \left (\frac {\sqrt {3} C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}, \ y{\left (t \right )} = C_{3} e^{t} - \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} + \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}, \ z{\left (t \right )} = C_{1} e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - C_{2} e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{3} e^{t}\right ]
\]