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1-a. For what values of k

style="color:rgb(0,0,0);background-color:rgb(255,255,255);"> does the function y = cos(kt) satisfy the differential equation 9y'' = −25y?

(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 = A sin(kt) + B cos(kt)

is also a solution.

y = A sin(kt) + B cos(kt)    ⇒     y' = Ak cos(kt) − Bk sin(kt)    ⇒     y'' = −Ak2 sin(kt) − Bk2 cos(kt).

The given differential equation 9y'' = −25y

is equivalent to 9y'' + 25y =?

thus, LHS = 9y'' + 25y= 9(−Ak2 sin(kt) − Bk2 cos(kt)) + 25( ? )

= 9Ak2 sin(kt) 9Bk2 cos(kt) +( ? )  sin(kt) + 25B cos(kt)

=(25 9k2)A sin(kt) +  ( ?) cos(kt)

=( ? )since k2 =( ?)

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