Question

Find a positive value of k for which y = \sin(k t) satisfies \frac{d^2y}{dt^2} + 16 y = 0.

Answer 1

Given:
y = sin(kt) satisfies the ODE
$\frac{d^{2}y}{dt^{2}} + 16y = 0$

Evaluate the derivatives of y.
y' = k cos(kt)
y'' = -k² sin(kt)

To satisfy the ODE requires that
-k² sin(kt) + 16 sin(kt) = 0

Either k² - 16 = 0 or sin(kt) = 0.
When k² - 16 = 0, obtan
k = 4   (for a positive value of k)
When sin(kt) = 0,
kt = nπ, for n=1,2,3, ...,

Answer:  k=4