(a) For what values of k does the function y = cos(kt) satisfy the differential equation 25y" = -9y? (Enter your answers as a comma-separated list.) 3 3 5'5 -(--) 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-Aksin (kt)- Bk2 cos(kt) Note that the given differential equation 25y" = -9y is equivalent to 25y" + 9y= 0 Now, substituting the expressions for y and y" above and simplifying, we have = = LHS 25y+9y 25-Ak²sin (kt) -Ak²sin (kt) - Bk² cos(kt) +9(A sin(kt) + B cos(kt)) --25 Aksin (kt) 25Bk2 cos(kt)+9A sin(kt) + 9B cos(kt) = - (9 25k2) sin(kr) )+ +(925k2) B cos(kt) = 0 since for all value of k found above, k² = 9 = 25
(a) For what values of k does the function y = cos(kt) satisfy the differential equation 25y" = -9y? (Enter your answers as a comma-separated list.) 3 3 5'5 -(--) 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-Aksin (kt)- Bk2 cos(kt) Note that the given differential equation 25y" = -9y is equivalent to 25y" + 9y= 0 Now, substituting the expressions for y and y" above and simplifying, we have = = LHS 25y+9y 25-Ak²sin (kt) -Ak²sin (kt) - Bk² cos(kt) +9(A sin(kt) + B cos(kt)) --25 Aksin (kt) 25Bk2 cos(kt)+9A sin(kt) + 9B cos(kt) = - (9 25k2) sin(kr) )+ +(925k2) B cos(kt) = 0 since for all value of k found above, k² = 9 = 25
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...
Related questions
Question
Transcribed Image Text:(a) For what values of k does the function y = cos(kt) satisfy the differential equation 25y" = -9y? (Enter your answers as a
comma-separated list.)
3 3
5'5
-(--)
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-Aksin (kt)- Bk2 cos(kt)
Note that the given differential equation 25y" = -9y is equivalent to 25y" + 9y= 0
Now, substituting the expressions for y and y" above and simplifying, we have
=
=
LHS 25y+9y 25-Ak²sin (kt)
-Ak²sin (kt) - Bk² cos(kt)
+9(A sin(kt) + B cos(kt))
--25 Aksin (kt)
25Bk2 cos(kt)+9A sin(kt) + 9B cos(kt)
=
-
(9 25k2) sin(kr)
)+
+(925k2) B cos(kt)
= 0
since for all value of k found above, k² =
9
= 25
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning