(a) for what values of k does the function y = cos(kt) satisfy the differential equation 16y'' = −81y?
1 answer:
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Answer:
4
Step-by-step explanation:
Given that
Now,
This can be written as
[as
]
[ by using equation (i)]
=4
Hence,
Answer:
Step-by-step explanation:
I used 3 of the coordinates to solve for the a, b, and c that we need in
We have 3 unknowns so we need 3 equations. The coordinates I used are from your table:
(-2, 12), (-1, 6), (0, 2). Start with (0, 2). Sub in a 2 for y and a 0 for x in the standard form of the quadratic:
which simplifies very nicely to
c = 2. We will use that value now when we do this again using (-2, 12):
and
12 = 4a - 2b + 2 and
4a - 2b = 10 (1)
Do it again with the third coordinate (-1, 6):
and
6 = 1a - b + 2 and
a - b = 4 (2)
Now we have a system of equations, (1) and (2) that we will combine and solve for a:
4a - 2b = 10
a - b = 4
Multiply the second equation by -2 to get a new system:
4a - 2b = 10
-2a + 2b = -8
Add to get
2a = 2 so
a = 1
Now sub in 1 for a in (2):
1 - b = 4 and
-b = 3 so
b = -3
Now we have all that we need to write the equation: