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Laplace transformation: system of DE

  1. May 16, 2014 #1
    1. The problem statement, all variables and given/known data
    Let ##y_1^{'}+y_1=y_2##, ##y_2^{'}+5y_2=y_3##, ##y_3^{'}+y_3=f## and ##y_1(0)=y_2(0)=y_3(0)=0##. Find ##Y_1(s)## in terms of ##F(s)##.



    2. Relevant equations



    3. The attempt at a solution

    I am completely lost here. I tried to rewrite the system so that I would somehow get rid of ##y_2## and ##y_3## but that didn't really work out well. I don't know what to do.
     
  2. jcsd
  3. May 16, 2014 #2

    Ray Vickson

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    Let ##Y_i(s)## be the Laplace transform of ##y_i(t),## for ##i = 1,2,3##. Develop and solve the equations for ##Y_1, Y_2, Y_3##. (That is one of the standard methods for dealing with constant-coefficient linear DE systems.)
     
    Last edited: May 16, 2014
  4. May 16, 2014 #3

    BiGyElLoWhAt

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    Looks to me like you have to follow down the line:
    if:
    ##y_{1}' +y_{1} = y_{2}##
    and ##y_{2}' + 5y_{2} = y_{3}##
    then via substitution:

    ##(y_{1}' + y_{1})' + 5(y_{1}' + y_{1}) = y_{1}'' + (y_{1}' + 5y_{1}') + y_{1} = y_{3}##
    follow this logic and I believe you will get your answer.
     
  5. May 16, 2014 #4
    Doing so brings me to:

    ##y_1^{'''}+7y_1^{''}+11y_1^{'}+5y_1=f##

    Now I am guessing that all that still has to be done is:
    ##Y_1^{'''}(s)+7Y_1^{''}(s)+11Y_1^{'}(s)+5Y_1(s)=F(s)##

    And that's it?
     
  6. May 16, 2014 #5

    BiGyElLoWhAt

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    Well, that's what I was thinking, but I didn't really pay much attention to your notation at the beginning, is
    ##y_{1}## really ##y_{1}(t)## ?
    If that's the case then I guess they want (as previously mentioned) the Laplace transform of ##y_{1}## in terms of the Laplace transform of F. (judging by F(s)).
    If by F you mean f and Y you mean y, then yes I would say that that answer would suffice.
     
  7. May 16, 2014 #6

    BiGyElLoWhAt

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    I'm only just learning about Laplace Transforms currently myself. Sorry if these posts weren't much help
     
  8. May 16, 2014 #7
    Hmmm,

    BiGyElLoWhAt, your idea works but not so fast. :D

    The problem is that ##f^{(n)}(t)=s^nF(s)-\sum_{k=1}^{n}s^{k-1}f^{(n-k)}(0)## but the problem says nothing about the value of ##y^{'}##...

    So like Ray said, I firstly have write each equation with laplace transformation and than apply your idea which brings me to:

    ##Y_1(s)[s^3+7s^2+11s+5]=F(s)##

    Is this the final result?
     
  9. May 16, 2014 #8

    LCKurtz

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    That is almost done (and correct). You haven't finished solving for ##Y_1## in terms of ##F## until you divide both sides by that polynomial in ##s##. If your ultimate goal is to finish solving by inverting the ##Y_i##, you would want to leave the polynomial in ##s## in factored form. Also don't you need to find the other ##Y_i\text{'s}##?
     
    Last edited: May 16, 2014
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