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Regular singular points of 2nd order ODE

  1. May 27, 2010 #1
    1. The problem statement, all variables and given/known data
    [PLAIN]http://img265.imageshack.us/img265/6778/complex.png [Broken]

    I did the coefficient of the w' term. What about the w term?

    This seems like a fairly standard thing, but I can't seem to find it anywhere.
    What ansatz should I use for q, if the eqn is written w''+pw'+qw?
    C/(z-a)²+ D/(z-b)²?
    Any conditions, except for the one generated by z->1/t substitution?
    Or should I use C+cz on the top, etc?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. May 27, 2010 #2

    gabbagabbahey

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    Well, if a 2nd order ODE has regular singular points at [itex]z=a[/itex] and [itex]z=b[/itex], then [itex]q[/itex] has poles up to 2nd order at those points, and the most general form of [itex]q[/itex] is then

    [tex]q(x)=\frac{g(z)}{(z-a)^2(z-b)^2}[/tex]

    where [itex]g(z)[/itex] is analytic everywhere. You should have used similar reasoning to find

    [tex]p(z)=\frac{f(z)}{(z-a)(z-b)}[/itex]

    Then just apply the linearity condition to find [itex]f[/itex] and [itex]g[/itex].
     
    Last edited by a moderator: May 4, 2017
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