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{\bf ADVANCED DYNAMICS --- PHY 4936}\\\vspace {2mm} 
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\noindent Solution 28: 
$$\ddot x + \omega_1\,x\ =\ 0\,,~~~~~~ \ddot y + \omega_1\,y\ =\ 0\,,$$
Solution with $x_0=y_0=0$;
\begin{eqnarray} \nonumber
 x(t) &=& A_1\,\sin(\omega_1t) \\ \nonumber
 y(t) &=& A_2\,\sin(\omega_2t)
\end{eqnarray}
Now, $x(t_1)=y(t_1)=0$ for some future time $t_1>0$ implies
\begin{eqnarray} \label{1}
  \omega_1\,t_1\ =\ n\,\pi\,,~~~~~~\omega_2\,t_1\ =\ m\,\pi
\end{eqnarray}
with $n$, $m$ integers $\ge 1$. With $n$ and $m$ being the smallest
integers, so that (\ref{1}) holds, the time becomes
$$ t_1\ =\ \frac{n\,\pi}{\omega_1}\ =\ \frac{m\,\pi}{\omega_2}\ .$$
 
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