Final answer:
To find a positive value of k for which y = sin(kt) satisfies d²y/dt² + 25y = 0, you can solve the differential equation by finding the second derivative of y and substituting it into the equation. By factoring out sin(kt) and solving for k, you can determine the positive value of k that satisfies the equation is k = 5.
Step-by-step explanation:
To find a positive value of k for which y = sin(kt) satisfies d²y/dt² + 25y = 0, we need to find the value of k that makes the equation true.
First, let's find the second derivative of y. Taking the derivative of y = sin(kt) twice, we get:
dy/dt = kcos(kt)
d²y/dt² = -k²sin(kt)
Then, substitute these derivatives into the differential equation:
-k²sin(kt) + 25sin(kt) = 0
Now, let's factor out sin(kt) from the equation:
sin(kt)(25 - k²) = 0
This equation is true when either sin(kt) = 0 or (25 - k²) = 0.
To find the positive value of k, we solve the equation 25 - k² = 0:
k² = 25
k = 5
Therefore, the positive value of k that satisfies the given differential equation is k = 5.