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2nd Order Linear ODE Problem

  1. Feb 3, 2013 #1
    1. The problem statement, all variables and given/known data
    Find general solution to:

    xy''+2y'+4xy=0

    2. Relevant equations

    Frobenius Method or Bessel's Equation

    3. The attempt at a solution

    I know how to get the roots for this problem (which are r1 = 0 & r2 = -1). But not I don't know what to do with these roots. I know that this is considered "Case 3", but I seriously don't know what to do from here.

    Furthermore, I have attempted to solve for y1(x) and came up with the answer:

    y1(x) = a0(x^-1)cos(2x) + ((a1)/2)(x^-1)sin(2x)

    If this is right, then I'm not sure what the next step is. Do I do reduction of order? I hope not because that would take FOREVER.

    Wolfram Alpha has the general solution as

    y= (c1(exp(-2ix)))/x - (ic2(exp(2ix)))/4x

    I'm totally lost. Any help would be appreciated.
     
  2. jcsd
  3. Feb 4, 2013 #2

    haruspex

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    Did you check that satisfies the original equation? (It doesn't look right to me. I get exponentials, not trig.)
     
  4. Feb 4, 2013 #3
    No I didn't, but exponentials match the wolfram alpha solution. Can you maybe explain how you got the exponentials?
     
  5. Feb 4, 2013 #4

    haruspex

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    The way it works on these fora is that you post your working and others try to spot where you went wrong.
     
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