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Laplace tranforms with boundary conditions

  1. Mar 29, 2012 #1
    1. The problem statement, all variables and given/known data
    Here's the question:

    Use laplace transforms to find X(t), Y(t) and Z(t) given that:

    X'+Y'=Y+Z
    Y'+Z'=X+Z
    X'+Z'=X+Y

    subject to the boundary conditions X(0)=2, Y(0)=-3,Z(0)=1.

    Now I have learnt the basics of laplace transforms, but have not seen a question in this form before. How do I start the question, could someone for instance show me how to get X(t) and I'll try the rest knowing how to do it? I have other questions I need to do like this, but this looks like the easiest one.

    Thanks


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 29, 2012 #2

    LCKurtz

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    Take the transform of each equation so instead of 3 equations in X(t), Y(t), and Z(t) you have 3 equations in their transforms x(s), y(s), and z(s). Then solve the 3 equations for those three transforms.
     
  4. Mar 29, 2012 #3
    I am still unsure.

    Take L() to be notation for laplace.

    Top line,

    L(X')=sx(s)-X(0)
    L(Y')=sy(s)-Y(0)
    L(Y)=y(s)
    L(Z)=z(s)

    How do I solve from here?

    sx(s)-X(0)+sy(s)-Y(0)=y(s)+z(s)
     
  5. Apr 2, 2012 #4

    LCKurtz

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    Sorry for the delay in getting back. Google has started intercepting Forum posts as spam and I didn't know it. What you need to do is take the LaPlace transform of both sides of all three equations, using formulas like you have listed above. You will get 3 equations in the three unknowns x(s), y(s), and z(s). And you know X(0) = 2 and Y(0)=-3 so use that. Plug the equations above into the first equation. Then do likewise with the other two.
     
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