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(a) For what values of k does the function y = cos(kt) satisfy the differential equation 4y"= -81y? (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. We begin by calculating the following. y = A sin(kt) + B cos(kt) y' = Ak cos(kt) - Bk sin(kt) y" = Note that the given differential equation 4y" = -81y is equivalent to 4y" + 81y = Now, substituting the expressions for y and y" above and simplifying, we have LHS = 4y" +81y = + 81(A sin(kt) + B cos(kt)) = -4 = (814k2) = 0 since for all value of k found above, k² = 4Bk² cos(kt) + 81A sin(kt) + 81B cos(kt) + (81 - 4k²) B cos(kt)

(a) For what values of k does the function y = cos(kt) satisfy the differential equation 4y"= -81y? (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. We begin by calculating the following. y = A sin(kt) + B cos(kt) y' = Ak cos(kt) - Bk sin(kt) y" = Note that the given differential equation 4y" = -81y is equivalent to 4y" + 81y = Now, substituting the expressions for y and y" above and simplifying, we have LHS = 4y" +81y = + 81(A sin(kt) + B cos(kt)) = -4 = (814k2) = 0 since for all value of k found above, k² = 4Bk² cos(kt) + 81A sin(kt) + 81B cos(kt) + (81 - 4k²) B cos(kt)

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 1CR
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(a) For what values of k does the function y = cos(kt) satisfy the differential equation 4y" = -81y? (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. We begin by calculating the following. y = A sin(kt) + B cos(kt) → y' = Ak cos(kt) - Bk sin(kt) → y" = Note that the given differential equation 4y" = -81y is equivalent to 4y" + 81y = Now, substituting the expressions for y and y" above and simplifying, we have LHS = 4y" + 81y = + 81(A sin(kt) + B cos(kt)) = = (81 - 4k²) = 0 since for all value of k found above, k² = Need Help? Read It 4Bk² cos(kt) + 81A sin(kt) + 81B cos(kt) + (81 - 4k²) B cos(kt)
Transcribed Image Text:(a) For what values of k does the function y = cos(kt) satisfy the differential equation 4y" = -81y? (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. We begin by calculating the following. y = A sin(kt) + B cos(kt) → y' = Ak cos(kt) - Bk sin(kt) → y" = Note that the given differential equation 4y" = -81y is equivalent to 4y" + 81y = Now, substituting the expressions for y and y" above and simplifying, we have LHS = 4y" + 81y = + 81(A sin(kt) + B cos(kt)) = = (81 - 4k²) = 0 since for all value of k found above, k² = Need Help? Read It 4Bk² cos(kt) + 81A sin(kt) + 81B cos(kt) + (81 - 4k²) B cos(kt)
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