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Differential equations, variation of parameters

  1. Feb 15, 2010 #1
    1. The problem statement, all variables and given/known data
    Using variation of parameters, find the general solutions of the differential equation


    2. Relevant equations
    y''' - 3''y + 3y' - y = et / t
    where et/t = g(t)


    3. The attempt at a solution
    I know how to solve these types of equations when its a second order, but I don't understand what to do for the particular solution since there are 3 solutions to the associated homogeneous equation, y1 = et, y2 = tet, y3 = t2et.
    Usually I would just take the 2 solutions and compute the Wronskian, then use the formula where it's -y1*integral([y2*g(t)]/W)dt + y2*integral([y1*g(t)]/W)dt.
    Since there are three solutions though, I don't understand how to solve it. My textbook uses a different method where they use something like v1'y1 + v2'y2 + v3'y3 = 0, v1'y1' + v2'y2' + v3'y3' = 0, and then the next equation is the same except the y's are the 2nd derivatives and it = g(t).
    Then they solve for v1, v2 and v3, integrate, and plug them into the homogeneous equation to get the particular solution.
    Sorry if this isn't clear!
     
  2. jcsd
  3. Apr 18, 2010 #2
    Last edited by a moderator: May 4, 2017
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