help with calculus?

Question

this is my second qs. Thanks to the person whow explain the first one to me.
here is the qs
find the value of w for which y = cos(wt) satisfies (d^2y/dt^2) + 4y= 0

I have the answer but i dont now how to do the problem.

Answer 1

OK, two main steps:

Step 1: find f''(t) of cos (wt):
Use the chain rule:
given f(g(x)), the derivative = f'(g(x)) * g'(x).
f(t) = cos t, g(t) = wt
f'(t) = -sin t, g'(t) = w
f'(g(x)) * g'(x)= -sin (wt) * w
thus: f''(t) = -cos (wt) * w^2 = -w^2 cos (wt).

Step 2: Use substituteion to solve -w^2 cos (wt) + 4y = 0 for w.
-w^2 cos (wt) + 4y = 0
-w^2 cos (wt) + 4 cos (wt) = 0 (substitute for y)
-w^2 + 4 = 0 (divide by cos (wt))
4 = w^2
2 = w (solution!)