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ODE with Fourier Series

  1. Oct 21, 2006 #1

    Ne0

    User Avatar

    Ok we are given the ODE
    [tex]
    {y}^{\prime\prime}(t) + \omega^2{y(t)} = {r(t)}
    [/tex]
    [tex]
    r(t) = cos\omega{t}
    [/tex]
    [tex] \omega = 0.5,0.8,1.1,1.5,5.0,10.0
    [/tex]
    I know you can use variation of paramaters to solve for it so I start by finding the complementary solution.
    [tex]
    {y}^{\prime\prime}(t) + \omega^2{y(t)} = 0
    [/tex]
    We know solutions are of the form
    [tex]
    y = \exp{(mt)}
    [/tex]
    so after taking derivatives and what not we get the fundamental solution
    [tex]
    \cos\omega{t}, \sin\omega{t}
    [/tex]
    Our complementary solution is
    [tex]
    {y}_{c}=Acos \omega{t} + Bsin \omega{t}
    [/tex]
    For the particular solution we set
    [tex]
    {y}^{\prime\prime}(t) + \omega^2{y(t)} = cos\omega{t}
    [/tex]
    We then use a fourier series to expand
    [tex]
    cos\omega{t}
    [/tex]
    Then proceed to solve for it but the problem I'm having is that I'm getting the fourier series to be zero which is strange. I know that there will be no
    [tex]
    {b}_{n}
    [/tex]
    term since cos is even but its still werid why I'm getting zero for
    [tex]
    {a}_{0}, {a}_{n}
    [/tex]
    Any help would be appreciated.
     
    Last edited: Oct 21, 2006
  2. jcsd
  3. Oct 21, 2006 #2

    Ne0

    User Avatar

    I swear this latex thing I can't figure it out.
     
  4. Oct 21, 2006 #3

    Ne0

    User Avatar

    If you guys are stuck the answer in the book is:
    [tex]
    {y} = {c}_{1}\cos\omega{t} + {c}_{2}\sin\omega{t} + {A}(\omega)\cos\omega{t}
    [/tex]

    [tex]
    {A}(\omega) = \frac{1}{\omega^2 - 1} {\leq} 0
    [/tex]
    if [tex] \omega^2 {\leq} 1 [/tex]

    [tex]
    {A}(\omega) = \frac{1}{\omega^2 - 1} {\geq} 0
    [/tex]
    if [tex] \omega^2 {\geq} 1 [/tex]

    Since there was not only greater then and less then I had to use less then or equal and greater then or equal
     
    Last edited: Oct 21, 2006
  5. Oct 21, 2006 #4

    Ne0

    User Avatar

    Can anyone help please with what I am doing wrong in the Fourier expansion?
     
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