Find a positive value of k for which y = \sin(k t) satisfies \frac{d^2y}{dt^2} + 16 y = 0.
1 answer:
Given:
y = sin(kt) satisfies the ODE
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
y = sin(kt) satisfies the ODE
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
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