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Homework Help: Second Order ODE - Variation of Parameters

  1. Mar 3, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the general solution of the following diff. eqn.

    y''(t) + 4y'(t) + 4y(t) = t^(-2)*e^(-2t) where t>0

    2. Relevant equations

    General soln - Φgeneral(t) + Φparticular(t)

    Wronskian - Φ1(t)Φ22'(t) - Φ2(t)Φ1'(t)

    3. The attempt at a solution

    I'm solving by variation of parameters.

    First solving for the general solution, y'' + 4y' + 4y = 0

    r2 + 4r + 4 which factors into (r+2)(r+2), so r = -2, -2.

    So the gen solution is y = c11e^(-2t) + c2e^(-2t)

    Now solving for the particular solution.

    Φ1 and Φ2= e^(-2t)

    The Wronskian here ends up being 2e^(-4t) - 2e^(-4t) which equals zero.

    What went wrong here? I know the Wronskian cannot equal zero here. This is where I am stuck.
  2. jcsd
  3. Mar 3, 2009 #2
    Wow, two seconds after I posted this I realize what I did wrong. Φ2 is equal to te^(-2t). Not e^(-2t). But that still equals zero haha.

    Edit again: I didn't use product rule. That was the problem.
    Last edited: Mar 3, 2009
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