QUESTION 1
For what values of $k$ does the function $y=\cos k t$ satisfy the differential equation $9 y^{\prime \prime}=-25 y$ ?

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 \(k = \boxed{-1.6666666666666667}\).