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Math Help - Step Functions

  1. #1
    Junior Member
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    Berkeley, California
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    Step Functions

    Hi all, I'm having a bit of trouble solving this problem. The instructions say to find the Laplace transform of the given function.

    The given function:

    f(t)=\left\{<br />
\begin{array}{l l}   <br />
      0,& t<2\\<br />
      (t-2)^3, & t \geq 2<br />
\end{array}\right.

    Phew! Trying to create that in LaTeX was probably just as difficult

    Thanks for reading!
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  2. #2
    MHF Contributor

    Joined
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    Quote Originally Posted by TheBerkeleyBoss View Post
    Hi all, I'm having a bit of trouble solving this problem. The instructions say to find the Laplace transform of the given function.

    The given function:

    f(t)=\left\{<br />
\begin{array}{l l}   <br />
      0,& t<2\\<br />
      (t-2)^3, & t \geq 2<br />
\end{array}\right.

    Phew! Trying to create that in LaTeX was probably just as difficult

    Thanks for reading!
    Well, by the definition of "Laplace Transform" this is \int_2^\infty e^{-st}(t- 2)^3 dt. That should be doable with integration by parts- three times.
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  3. #3
    Member
    Joined
    Aug 2009
    From
    Mumbai
    Posts
    83
    You can solve this by redefining the function in terms of the unit step function u(t)

    u2(t)=0 t<2
    u2(t)=1 t>=2

    So the given function may be written as

    f(t)=u2(t)(t-2)^3


    Now, an important property regarding laplace transforms of unit step functions is

    L(ua(t)g(t-a)) = e^{-as}L(g(t))

    Therefore here, g(t)=t^3 and a=2

    So the laplace transform can be evaluated using the above property
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