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Identifying a second-order ODE

  1. Aug 7, 2012 #1
    Is there a general solution to

    [tex] \frac{d}{dt}\left[p(t)\frac{dx(t)}{dt}\right] + q(t)x(t) = 0 [/tex]

    for [itex] x(t) [/itex] when [itex] p(t) [/itex] and [itex] q(t) [/itex] are arbitrary functions? Better yet, does this question have a name, or some identifier, that I could look in to? It might appear more familiar written as

    [tex] \left[p(t)x^\prime\right]^\prime + q(t)x = 0 [/tex]
     
  2. jcsd
  3. Aug 8, 2012 #2
    No, there is no genertal solution
    .
     

    Attached Files:

  4. Aug 10, 2012 #3
    Look up Sturm-Liouville problems or equations.
     
  5. Aug 11, 2012 #4
    I think that's just a ODE with non constant coeffecients, since expanding yields

    P(t)x''+P'(t)x'+q(t)x=0

    You may be able to solve this with power series if P and q fit them.
    Non linear differential equations rarely have closed form solutions.
    But that's okay, we have computers
     
  6. Aug 11, 2012 #5
    Of course, when I say "There is no general solution", I mean "No general analytical solution espressed on a closed form".
    Obviously, in some particular cases, with some particular forms of functions p(t) and q(t), the solutions might be known on closed form, and/or be expressed as infinite series.
    Even more generally the solutions can be accurately approached thanks to numerical methods.
     
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