Step 1 :The given differential equation is a second order linear homogeneous differential equation. The general solution of such an equation is given by \(y(t) = A \cos(\sqrt{\frac{25}{9}}t) + B \sin(\sqrt{\frac{25}{9}}t)\), where A and B are constants.
Step 2 :In the function \(y=\cos kt\), the coefficient of \(t\) in the cosine function is \(k\). Comparing this with the general solution, we can see that \(k\) must be equal to \(\sqrt{\frac{25}{9}}\) or \(-\sqrt{\frac{25}{9}}\) for the function to satisfy the differential equation.
Step 3 :Final Answer: The values of \(k\) that satisfy the differential equation are \(k = \boxed{1.6666666666666667}\) and \(k = \boxed{-1.6666666666666667}\).