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

  1. Feb 18, 2013 #1
    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...
     
  2. jcsd
  3. Feb 18, 2013 #2

    Ray Vickson

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    Well, do you have 4y'' = -25y or not?
     
  4. Feb 18, 2013 #3
    I believe you would if y''=-Ak2sinkt-Bk2coskt?
     
  5. Feb 18, 2013 #4

    HallsofIvy

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    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.
     
  6. Feb 18, 2013 #5
    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
     
    Last edited: Feb 18, 2013
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