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The second shifting theorem and the unit step function

  1. Nov 6, 2011 #1
    1. The problem statement, all variables and given/known data

    I am trying to do some revision for an upcoming exam and one question I am trying to figure out is

    Use the second shifting theorem to find the Laplace transfrom of the following function:
    f(t) = t2, t < 4
    t, t ≥ 4

    2. Relevant equations



    3. The attempt at a solution
    I just don't understand how to get from the question to
    f(t) = t2[1 - u(t-4)] + tu(t-4)
    I am really struggling with applying the second shifting theorem to express in terms of the unit step function I am failing to see how it works because nothing is explained in basic detail?
     
  2. jcsd
  3. Nov 6, 2011 #2

    LCKurtz

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    Think of it this way. You start out with f(t) = t2. Then at t = 4 you want to take out the t2 and put in t, so you add the term u(t-4)(-t2+t).

    Then put it all together:

    f(t) = t2+u(t-4)(-t2+t)

    Now, if you wish, you can collect terms on the various powers of t:

    f(t) = t2(1-u(t-4)) + tu(t-4)
     
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