For what values of k does the function y = cos kt satisfy the differential equation 4y'' = - 9y? (enter your answers as a comma-separated list.)

Differentiating the function two times with respect to t:

y = cos (k x t)

y' = - k x sin (k x t)

y'' = - (k^2) x cos (k x t)

Plug this back into the Differential equation:

4 * (k^2) * cos (k*t) = - 9*cos (k*t)

4*k^2 = 9

k^2 = 9/4

k = + 9/4 and k = - 9/4

k = + sqrt (9/4) and l = - sqrt (9/4)

k = 1.5 and I = - 1.5