$[2\mathrm{.}26/5\mathrm{.}26$ Points$]$

SCalcET$99\mathrm{.}1\mathrm{.}016.$

$($a$)$ For what values of $k$ does the function $y=cos(kt)$ satisfy the differential equation $25y^{\text{'}\text{'}}=-81y?($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 $25y^{\text{'}\text{'}}=-81y$ is equivalent to $25y^{\text{'}\text{'}}81y=$

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

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

$q,$