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Math Help - Convolution of a DE

  1. #1
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    Convolution of a DE

    Hi,

    I don't even know where or how to start this problem and really need help.

    The problem is:
    Consider the differential equation
    Find a function
    , so that the particular solutions is given by the convolution

    I don't know what the question is really asking and how to approach it.

    Can someone please help

    Thanks!


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  2. #2
    MHF Contributor
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    Re: Convolution of a DE

    Quote Originally Posted by JC05 View Post
    Hi,

    I don't even know where or how to start this problem and really need help.

    The problem is:
    Consider the differential equation
    Find a function
    , so that the particular solutions is given by the convolution
    [SIZE=2]
    I don't know what the question is really asking and how to approach it.
    Do you know Laplace transforms and how to use them to solve a DE?

    taking Laplace transforms of both sides

    $(s^2+4s+13)X(s) = F(s)$

    $X(s)=\dfrac {F(s)}{s^2+4s+13}=\Theta(s) F(s)$ where

    $\Theta(s)=\dfrac 1 {s^2+4s+13}$

    multiplication in the $s$ domain corresponds to convolution in the $t$ domain so

    $\Theta(s)F(s) \Longleftrightarrow \theta(t) * f(t)$

    and thus

    $\theta(t) = \mathscr{L}^{-1} \{ \Theta(s) \} = \mathscr{L}^{-1} \left\{\dfrac 1 {s^2+4s+13} \right\}$
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