66.3.3 problem Problem 4
Internal
problem
ID
[13946]
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
:
Tuesday, January 28, 2025 at 06:09:21 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*}
✓ Solution by Maple
Time used: 0.070 (sec). Leaf size: 175
dsolve([diff(x(t),t)=y(t),diff(y(t),t)=z(t),diff(z(t),t)=x(t)],singsol=all)
\begin{align*}
x \left (t \right ) &= c_{1} {\mathrm e}^{t}+{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +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 {{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}+\frac {{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{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 {{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{-\frac {t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2}}{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*}
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 234
DSolve[{D[x[t],t]==y[t],D[y[t],t]==z[t],D[z[t],t]==x[t]},{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*}