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Diff Eq and the Dirac Delta(impulse) function.

  1. Jan 26, 2005 #1
    Ok, I was given: Solve the following using superposition:

    [tex]\ddot{x}+2\dot{x}+4x=\delta(t)[/tex]

    bounded by [tex]\dot{x}=0, x(0)=0[/tex]

    I solved the Homo eqn and got the following:

    [tex]x(t)=e^{-t}(\cos (\sqrt{3}t)+\frac{\sqrt{3}}{3}\sin ({\sqrt{3}t))[/tex]

    I also know that :
    [tex]\ddot{x}+2\dot{x}+4x=u(t)[/tex]

    equals

    [tex]x(t)=e^{-t}(\cos (\sqrt{3}t)+\frac{\sqrt{3}}{3}\sin ({\sqrt{3}t))+\frac{1}{4}[/tex] from a previous problem.

    So, I said:

    [tex]x(t)=e^{-t}(\cos (\sqrt{3}t)+\frac{\sqrt{3}}{3}\sin ({\sqrt{3}t))+\frac{\delta (t)}{4}[/tex]

    Is this correct thus far (the superposition part at least I'm pretty sure the diff eq portion is correct).

    How do I deal with the delta function? Any help would be much appreciated!!!
     
    Last edited: Jan 26, 2005
  2. jcsd
  3. Jan 26, 2005 #2

    dextercioby

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    Yes,u need to find the Green function...Do u know how to use the theorem of residues...??

    Daniel.
     
  4. Jan 26, 2005 #3
    Nope. My professor flew through this the other day (a similar problem) and looking at my notes it seems I missed something while he was talking.

    edit:

    Should I use the Homo eqn and the characteristic eqn for a delta function and solve using variation of parameters?

    edit:edit:
    I can't use the above because I only have two initial eqn's correct?
     
    Last edited: Jan 26, 2005
  5. Jan 26, 2005 #4

    dextercioby

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    I don't know how to do it otherwise...It's a LINEAR second order nonhomogenous ODE and you're asked for the Green function of the LINEAR DIFFERENTIAL OPERATOR
    [tex] \hat{O}=:\frac{d^{2}}{dt^{2}}+2\frac{d}{dt}+4 [/tex]

    Maybe someone else could come up with a different solution which wouldn't require distributions and residues theorem.

    Daniel.
     
  6. Jan 26, 2005 #5
    Anyone else out there in physics land have a hint for me?
     
  7. Jan 26, 2005 #6

    Hurkyl

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    Hrm, I can see two other approaches that might work.

    (1) Build the solution in patches. Note that if you restrict the domain to t >= 0, or t <= 0, the differential equation is purely homogenous. Then, the goal is to pick solutions on each of these domains that, when patched together, satisfy the differential equation at 0.

    (2) Write &delta;(t) as a limit of something, maybe as a limit of step functions. Then, with some hand-waving and some optimism, you can take the limit of the corresponding solutions to get a solution to your desired equation.
     
  8. Jan 26, 2005 #7
    Thanks!!! So simple... I used approach number 1 where time is defined t>=0 so there is only one instant where the delta function applies t=0.

    I have:
    [tex]x(t)=e^{-t}(\cos (\sqrt{3}t)+\frac{\sqrt{3}}{3}\sin ({\sqrt{3}t))[/tex]

    with my interval greater than t=0 as my answer.

    Thanks for the help.
     
  9. Jan 26, 2005 #8
    Here are a couple of links that will give you the basics about the delta function.

    http://mathworld.wolfram.com/DeltaFunction.html

    http://en.wikipedia.org/wiki/Dirac_delta

    Hopefully you wont get lost and should find that info relevent and useful. They cover the basic standard definitions plus some more of the delta function.

    One thig I noticed off the bat was that your solution did not match your contstraint of x(0)=0. If that boundry was given, then your solution should be zero at t=0. Also it is an inhomogenous ODE, so solve it as such.

    If you are so inclined and have the time, here is a link to from mathworld on Green's functions, you will be seeing them in the future.

    http://mathworld.wolfram.com/GreensFunction.html

    Enjoy! :biggrin:
     
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