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Solve y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms, g(t) is given

  1. Mar 22, 2012 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Solve the differential equation y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms.

    The g(t) function is a piecewise function and is attached as g.jpg.


    2. Relevant equations
    Laplace transforms of regular and unit step/Heaviside functions.


    3. The attempt at a solution
    My work is attached as MyWork.jpg. I suspect the part with the e to be the culprit but I'm not specifically sure as to what I did wrong and would appreciate it if someone could point it out to me.

    Thanks in advance!
     

    Attached Files:

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    • MyWork.jpg
      MyWork.jpg
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  2. jcsd
  3. Mar 23, 2012 #2

    LCKurtz

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    Re: Solve y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms, g(t) is give

    I didn't check all your steps, but it looks to me you haven't taken into account the f(t) = t part. Your formula for that is$$
    f(t) = t(1-u(8\pi))+8\pi u(t-8\pi)$$
     
  4. Mar 23, 2012 #3

    s3a

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    Re: Solve y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms, g(t) is give

    What does u(8π) mean?
     
  5. Mar 23, 2012 #4

    Ray Vickson

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    Re: Solve y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms, g(t) is give

    Why don't you just use the definition, rather than applying formulas that can be misused (as you did)? For [itex]g(t) = \min(t,8 \pi)[/itex] we have
    [tex] L[g](s) = \int_0^\infty e^{-st} g(t) \, dt = \int_0^{8 \pi} e^{-st} t \, dt
    + \int_{8 \pi}^\infty e^{-st} 8 \pi \, dt, [/tex] and just do both integrals.

    RGV
     
    Last edited: Mar 23, 2012
  6. Mar 23, 2012 #5

    LCKurtz

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    Re: Solve y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms, g(t) is give

    Sorry, that was a typo; that expression should be ##u(t-8\pi)## but the time for me to edit and correct it has expired.
     
  7. Mar 23, 2012 #6
    Re: Solve y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms, g(t) is give

    u(t-8∏) is a unit step function. You can think of it as energy going into your system after 8∏ time has elapsed. The function is 0 <= 8∏ and 1 after 8∏.
     
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