$(a)$ For what values $ofk$ does the function $y=cos(kt)$ satisfy the differential equation $25y^{\text{'}\text{'}}=-81y?(Enter$ your answers $as$ a comma$-$separated list.$)$

$k=\frac{9}{5},-\frac{9}{5}$

$(b)$ For those values $ofk,$ 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 $25y^{\text{'}\text{'}}=-81yis$ equivalent $to25y^{\text{'}\text{'}}+81y=$

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

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