$($a$)$ For what values of $k$ does the function $y=cos(kt)$ satisfy the differential equation $49$y$*=-9y?($Enter your answers as a comma$-$separated list.$)k=$

$($b$)$ For those values of $k,$ verify that every member of the family of functions $y=$Asin$(kt)$ Bcos$(kt)$ is also a solution. We begin by calculating the following.

$y=$Asin$(kt)$ Bcos$(kt)=>y^{\text{'}}=$Akcos$(kt)-$Bksin$(kt)=>y^{\text{'}\text{'}}=$

$$

Note that the given differential equation $49y^{**}=-9y$ is equivalent to $49y^{**}9y=$

Now, substituting the expressions for $y$ and $y^{**}$ above and simplifying, we have

USS $=49y9y=49()9(Asin(kt)$ Bcos$(kt))$

$=-49()49B{k}^{2}cos(kt)9$Asin$(kt)9$Bcos$(kt)$

$=(9-49k^2)(-49k^2)Bcos(kt)$

$=0$

since for all value of $k$ found above, $k^2=$