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Laplace transform for solving ODE with variable coefficients

  1. Dec 4, 2006 #1
    Can we use laplace transform to solve an ODE with variable coefficients?

    Like this one:

    4x y" + 2 y' + y = exp (-x)
     
  2. jcsd
  3. Dec 4, 2006 #2

    HallsofIvy

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    Staff Emeritus
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    Theoretically, yes, if you can find the Laplace transform of f(x)y"(x)!

    The Laplace transform of xy"(x) is, by definition,
    [tex]\int_0^\infty e^{-st}ty"(t)dt[/tex]
    Integrate by parts with [itex]u= e^{-st}t[/itex] so that [itex]du= (e^{-st}- st e^{-st})dt[/itex], dv= y"(t)dt so that v= y'(t). Then
    [tex]-\int_0^\infty (e^{-st}- ste^{-st})y'(t)dt[/tex]
    Do that by integration by parts with [itex]u= (e^{-st}- ste^{-st}[/itex] so that [itex]du= -2e^{-st}+ s^2te^{-st}[/itex], dv= y'(t)dt so that v= y. Then
    [tex]2y(0)+ \int_0^\infty (2e^{-st}- s^2te^{-st})y(t)dt[/itex]
    Write that last as a Laplace transform of y and reduce to an algebraic equation as usual.

    Unfortunately, that is not always easy to do with general variable coefficients. The Laplace transform is basically a method for very mechanically solving linear equations with constant coefficients.
     
  4. Dec 4, 2006 #3
     
    Last edited: Dec 4, 2006
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