66.3.3 problem Problem 4

Internal problem ID [13875]
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 : Monday, March 31, 2025 at 08:15:54 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: 0.036 (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: 0.258 (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)
 

\[ \left [ x{\left (t \right )} = C_{3} e^{t} - \left (\frac {C_{1}}{2} + \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - \left (\frac {\sqrt {3} C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}, \ y{\left (t \right )} = C_{3} e^{t} - \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} + \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}, \ z{\left (t \right )} = C_{1} e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - C_{2} e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{3} e^{t}\right ] \]