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Verify every number of a family of functions is also a solution

by Aerospace93
Tags: family, functions, number, solution, verify
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Aerospace93
#1
Feb18-13, 11:35 AM
P: 22
1. The problem statement, all variables and given/known data
(a)For what values of k does the function y=coskt satisfy the differential equation 4y''=-25y?
(b) For those values of k, verify that every number of the family of functions y=Asinkt+Bcoskt is also a solution.

3. The attempt at a solution
(a) y=coskt, y'=-ksinkt, y''=-k2coskt.
4(-k2coskt)=-25coskt; 4k2=25, so k=+/-5/2
(b) y= Asinkt+Bcoskt; y'= +/-Akcoskt+/-BKcoskt; y''= +-Ak2sinkt+/-bk2coskt
Im not sure how to verify this...
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Ray Vickson
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Feb18-13, 11:42 AM
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Quote Quote by Aerospace93 View Post
1. The problem statement, all variables and given/known data
(a)For what values of k does the function y=coskt satisfy the differential equation 4y''=-25y?
(b) For those values of k, verify that every number of the family of functions y=Asinkt+Bcoskt is also a solution.

3. The attempt at a solution
(a) y=coskt, y'=-ksinkt, y''=-k2coskt.
4(-k2coskt)=-25coskt; 4k2=25, so k=+/-5/2
(b) y= Asinkt+Bcoskt; y'= +/-Akcoskt+/-BKcoskt; y''= +-Ak2sinkt+/-bk2coskt
Im not sure how to verify this...
Well, do you have 4y'' = -25y or not?
Aerospace93
#3
Feb18-13, 11:49 AM
P: 22
I believe you would if y''=-Ak2sinkt-Bk2coskt?

HallsofIvy
#4
Feb18-13, 11:55 AM
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Verify every number of a family of functions is also a solution

Did you miss where the problem says "for those values of k"? You are asked to show that y= A sin((2/5)t)+ B cos((2/5)t) and y= Asin((-2/5)t)+ Bcos((-2/5)t) both satisfy [itex]4d^2y/dx^2= -25y[/quote] for any A and B.
Aerospace93
#5
Feb18-13, 12:37 PM
P: 22
ok so (for a positive k) y= Asinkt+Bcoskt; y'= -Akcoskt+BKsinkt; y''= -Ak2sinkt-bk2coskt
so,
-4[A(5/2)2sin(5/2)t-B(5/2)2cos(5/2)t= -25[Asin(5/2)t+Bcos(5/2)t
-25Asin(5/2)t-25Bcos/5/2)t= RHS


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