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Variation of Parameters

  1. Jul 9, 2011 #1
    Given t^2 y'' -t(t+2)y' = (t+2)y= 2t^3 and y1= t, y2= te^t
    Find the particular solution-

    I ve worked the problem to [ -2t^2 -2t] by:
    -t * Integral [ 2t* te^t/ t^2e^t] + te^t * Integral [ 2t^2/ t^2e^t]


    whereas the book states that it is simply -2t^2. Can you guys tell me where I made my mistake, I am stumped.
     
  2. jcsd
  3. Jul 11, 2011 #2
    Is that first = supposed to be +?
     
  4. Jul 11, 2011 #3
    To be honest, I don't completely understand this, but I'm not sure you've actually done anything wrong. You got the particular solution -2t^2-2t, and the book says -2t^2. But notice that one of the homogeneous solutions is t. If you add a linear combination of the homogeneous solutions to the particular solution, you get another equally valid particular solution. So, if -2t^2 -2t works, so does -2t^2 -2t + 2t = -2t^2.
     
  5. Jul 15, 2011 #4

    MathematicalPhysicist

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    Gold Member

    Well a private solution here will be of the form:
    At^3+Bt^2+Ct
    where you can see quite immediately that A=0 (cause it has a term of t^4 which obviously gets nullified).

    I did the calculation and got exactly as the book, try again.
    PS
    the C term doesn't get cancelled though.
     
    Last edited: Jul 15, 2011
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