$(a)$ For what values $ofk$ does the function $y=cos(kt)$ satisfy the differential equation $4y^{\text{'}\text{'}}=-81y?(Enter$ your answers $asa$

comma$-$separated list.$)$

$k=$

$(b)$ 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 $4y^{\text{'}\text{'}}=-81yis$ equivalent $to4y^{\text{'}\text{'}}+81y=$

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

$LHS=4y^{\text{'}\text{'}}+81y=4()+81(Asin(kt)+$Bcos$(kt))$

$=-4()-4B{k}^{2}cos(kt)+81$Asin$(kt)+81$Bcos$(kt)$

$=(81-4k^2)(,)+(81-4k^2)Bcos(kt)$

$=0$

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