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How do I solve: p(t)f''(t)+q(t)f'(t)=Kf(t) ?

  1. Mar 8, 2009 #1
    How do I solve the equation: p(t)f''(t)+q(t)f'(t)=Kf(t) ?

    I have the equation: p(t)f''(t)+q(t)f'(t)=Kf(t) that results from a problem of physics.
    - p(t) and q(t) are two known periodical functions of period T
    - f(t) is an unknown function
    - K is an unknown constant

    What I need is to determine all the possible values of these K constants with the intention of finding the real ones if they exists.
  2. jcsd
  3. Mar 9, 2009 #2
    Have you tried Fourier series like this?

    [tex]f(t) = \sum {a_n cos(\frac{n\pi t}{T})+b_n sin(\frac{n\pi t}{T})[/tex]

    Note that the fundamental frequency of the above series is half of that of p(t) and q(t).
  4. Mar 10, 2009 #3
    If we let y(t)=f(t) then your equation is a linear homogeneous DE.
    p(t)y''+q(t)y' - Ky = 0.

    But you are not particularly interested with the solution f(t), aren't you ?
    The existence theorem may be useful here I think! What is the initial condition?
  5. Mar 10, 2009 #4
  6. Mar 10, 2009 #5
    Out of curiousity, where's the physics? This is math.
  7. Mar 10, 2009 #6
    Flatmaster is right. Complex exponential form is better. And I think whatever k's value is, there is a general solution of the form

    [tex]f(t)=\sum{a_n e^{(r_n+in\omega)t}}[/tex]

    where [tex]\omega=\pi/T[/tex].

    For [tex]r_n<0[/tex], those components will die down. If there are [tex]r_n>0[/tex], the system will be unstable as those components will increase exponentially. For frequency components with [tex]r_n=0[/tex], they will be the steady state solutions. To be stable, k can assume a large range of values. To have steady state solutions, k can only assume discrete values. For those k's any slight change in k or p or q may render the system unstable, unless your system is non-linear beyond certain amplitude.

    P.S. My username is wywong, not wywrong B:).
  8. Mar 14, 2009 #7
    Thank you. Your idea with using Fourier series was good. Finally I investigated this method (reading documentation on the internet) and I come up with continuous domains of K for which the equality p(t)f''(t)+q(t)f'(t)=Kf(t) can take place.

    For instance if:
    K=[0 1] -> f(t) exists and it is not zero
    K=(1 4] -> f(t) does not exists
    K=(4-8] -> f(t) exists and it is not zero
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