Question

For what values of k does the function y i cos(kt) satisfy the differential equation 25y ∗ =−4y ?

Answer 1

The value of the constant k for which the function y = cos(kt) satisfies the differential equation 25y* = -4y is ±2/5, determined through differentiation and substitution.

Step-by-step explanation:

The student is asking about the determination of the constant k for which the function y = cos(kt) satisfies the differential equation 25y* = -4y. We must find the value of k that makes the equation true when y = cos(kt) is substituted into it. In the context of differential equations, when we look to find such a constant, we are typically looking for an angular frequency of a harmonic oscillator that causes the function to be a solution to the equation.

Now, let's take the second derivative of y = cos(kt) with respect to t:

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• The first derivative is -k sin(kt)
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• The second derivative is -k2 cos(kt)

Substitute y and its second derivative into the given differential equation:

25(-k2 cos(kt)) = -4 cos(kt)

If we simplify it, we get:

25k2 = 4

So:

k2 = 4/25

And k would be:

k = ±2/5

Therefore, the values of k that satisfy the differential equation are ±2/5.