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

in the attachment.   i didn't see your choices.   the word which means there is choices.

step-by-step explanation:

(x,y)

x tells you how much left or right you go from the origin

y tells you how much down or up you go from the origin

so for example (-3,2), you go left 3 and up twice. i denoted which of the ordered pairs is which in my graph.

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional