Internal problem ID [12210]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )\\ y^{\prime }\left (t \right )&=z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 176
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}+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*}
✓ Solution by Mathematica
Time used: 0.063 (sec). Leaf size: 234
DSolve[{x'[t]==y[t],y'[t]==z[t],z'[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*}