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Solve 3rd order ode using variation of parameters

  1. Nov 9, 2011 #1
    1. The problem statement, all variables and given/known data

    Solve using variation of parameters
    y''' - 2y'' - y' + 2y = exp(4t)


    2. Relevant equations
    Solve using variation of parameters


    3. The attempt at a solution

    I got the homogenous solutions to be 1, -1, and 2.

    So, y = Aexp(t) + Bexp(-t) + Cexp(2t) + g(t)

    I got W{exp(t), exp(-t), exp(2t)} = 6exp(2t)


    My professor did quite an unsatisfactory job explaining variation of parameter for high order equations (spent an hour and 30 minutes trying to do 1 problem, and still didn't finish it correctly). So I just need to be pointed in the right direction.

    Thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 10, 2011 #2

    CompuChip

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    Actually, I got -6 exp(2t), but maybe I made a sign error in my calculation.

    So the general idea is now to replace the ith column in the matrix by (0, 0, exp(4t)) and calculate the determinant |Wi| of that, for i = 1, 2, 3.
    Then if you calculate
    [tex]c_i(t) = \int \frac{|W_i|}{|W|} \, dt[/tex]
    the general solution is
    [tex]y(t) = c_1(t) e^{t} + c_2(t) e^{-t} + c_3(t) e^{-4t}[/tex]
     
  4. Nov 10, 2011 #3

    LCKurtz

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    As a learning exercise, fine. But I assume you know nobody in their right mind would use variation of parameters to find a particular solution, eh?
     
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