(a) For what values of k does the function y = cos(kt) satisfy the differential equation 100y" = -9y? (Enter your answers as a comma-separated list.) k = (b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution. y = A sin(kt) + B cos(kt) y' = Ak cos(kt) – Bk sin(kt) y" = -Ak2 sin(kt) – Bk2 cos(kt). The given differential equation 100y" = -9y is equivalent to 100y" + 9y = Thus, %3D LHS = 100y" + 9y= 100(-Ak² sin(kt) – Bk2 cos(kt)) + 9| -100AK2 sin(kt) – 100BK2 cos(kt) + |sin(kt) + 9B cos(kt) (9 – 100k2)A sin(kt) + |cos(kt) since k2 = %D
(a) For what values of k does the function y = cos(kt) satisfy the differential equation 100y" = -9y? (Enter your answers as a comma-separated list.) k = (b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution. y = A sin(kt) + B cos(kt) y' = Ak cos(kt) – Bk sin(kt) y" = -Ak2 sin(kt) – Bk2 cos(kt). The given differential equation 100y" = -9y is equivalent to 100y" + 9y = Thus, %3D LHS = 100y" + 9y= 100(-Ak² sin(kt) – Bk2 cos(kt)) + 9| -100AK2 sin(kt) – 100BK2 cos(kt) + |sin(kt) + 9B cos(kt) (9 – 100k2)A sin(kt) + |cos(kt) since k2 = %D
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Topic Video
Question
100%
Practice questions.
![(a) For what values of k does the function y =
cos(kt) satisfy the differential equation 100y" = -9y? (Enter your answers as a comma-separated list.)
k =
(b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution.
y = A sin(kt) + B cos(kt)
y' = Ak cos(kt) – Bk sin(kt)
y" = -Ak2 sin(kt) – Bk2 cos(kt).
The given differential equation 100y" = -9y is equivalent to 100y" + 9y =
Thus,
%3D
LHS =
100y" + 9y= 100(-Ak² sin(kt) – Bk2 cos(kt)) + 9|
-100AK2 sin(kt) – 100BK2 cos(kt) +
|sin(kt) + 9B cos(kt)
(9 – 100k2)A sin(kt) +
|cos(kt)
since k2 =
%D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F035c2915-f7cd-4b41-910e-788c320f285d%2Fa7072889-a308-45a6-9069-fb7f2a7740d4%2Ff1svuv7.png&w=3840&q=75)
Transcribed Image Text:(a) For what values of k does the function y =
cos(kt) satisfy the differential equation 100y" = -9y? (Enter your answers as a comma-separated list.)
k =
(b) For those values of k, verify that every member of the family of functions y = A sin(kt) + B cos(kt) is also a solution.
y = A sin(kt) + B cos(kt)
y' = Ak cos(kt) – Bk sin(kt)
y" = -Ak2 sin(kt) – Bk2 cos(kt).
The given differential equation 100y" = -9y is equivalent to 100y" + 9y =
Thus,
%3D
LHS =
100y" + 9y= 100(-Ak² sin(kt) – Bk2 cos(kt)) + 9|
-100AK2 sin(kt) – 100BK2 cos(kt) +
|sin(kt) + 9B cos(kt)
(9 – 100k2)A sin(kt) +
|cos(kt)
since k2 =
%D
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