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Help solving ODE's using eigenfunction expansion, in general

  1. Aug 11, 2012 #1
    My ODE textbook does not help me much here; and a large number of master's exam practice tests (with worked out solutions) also isn't helping me. I need someone to recommend a (preferably ONLINE) source that clearly states how to solves ODES using eigenfunction expansion.
    For example, y''+y=kcosx with y'(0)=0, y'(pie)=1.
    This is just an example of the type I want to learn how to solve.
    It seems that when the boundary conditions are homogeneous, this problem is easily solved by plugging in an infinite series: y=Sum(a_n*cos(nx)) into the left side, and then plugging in a different series: Sum(b_n*cosnx)=kcosx..then equating coefficients and getting the b_n's. BUT when the BC are non-homogeneous, there appears to be extra work.

    Help.
     
  2. jcsd
  3. Aug 12, 2012 #2
    Uhm... i am not sure you have a proper textbook.
    Also, I am not sure if you are familiar with fourier tranformation. It is an interesting way to deal a wide class of equations.
    Basically, it is almost the same thing you did but applying the transformation at both sides . You can easily look in the web. for exemple your example will became after transforming, something like..

    [itex] (1-w^2)\tilde{y}(w) = kC (\delta(w-1) + \delta(w+1)) [/itex]

    You see that the solution is straightforward as there is no derivatives. You should now inverse transform the solution if you want. I used some properties of Fourier transform to deal with the derivatives but again i think you should find it easily.
     
  4. Aug 13, 2012 #3
    Ok let me rephrase: can someone give me and work out a complete example with non-homogeneous boundary conditions, using eigenfunction expansion? let's say, y''+y=x+ cosx,
    y(0)=0, y(pie)=A..just to come up with one off the top of my head.
     
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