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

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