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Laplace Differential equation

  1. Aug 28, 2010 #1
    I have a differential equation that has to be solved with Laplace. I wish someone can provide a full answer

    y'' + 4y = x , 0<=x<π
    y'' + 4y = πe^-x , π<=x

    Initial Conditions:
    y(0)=0 y'(0)=1
  2. jcsd
  3. Aug 28, 2010 #2


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    Sorry we do not do that here.

    Start by applying Laplace tranforms to each of the equations.
  4. Aug 28, 2010 #3
    Learn this: http://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx
    and you will know what to do :)

    also your ODE can be written using the step (Heaviside) function:

    y'' + 4y = x + H[x-π](πe^(-x) - x)

    H[x-π] = 0 at x<π; H[x-π] = 1 at x>=π;

    Good Luck.
  5. Aug 28, 2010 #4


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    No "full answer" but:
    1) Solve y"+ 4y= x, 0<= x< [itex]\pi[/itex]
    with initial conditions y(0)= 0, y'(0)= 1.

    Evaluate the function, [itex]y_1(x)[/itex], satifying those conditions and its derivative at [itex]x= \pi[/itex] and solve
    2) [itex]y''[/itex][itex]+ 4y= [/itex][itex]\pi e^{-x}[/itex]
    with initial conditions [itex]y(\pi)= y_1(\pi)[/itex], [itex]y'(\pi)= y_1'(\pi)[/itex].
  6. Aug 29, 2010 #5
    Where I am stuck is how to transform the right part as to write it for the proper laplace transform
    How I would do it(and correct me where I am wrong)
    y'' + 4y = x[u(x-0)-u(x-π)] + πe^(-x)*u(x-π)

    How do you apply the Heavyside? Can you explain me your technique?
  7. Aug 29, 2010 #6
    Why would you want to use a Heaviside step function? Please advise.

    You need to transform the equation. What have you got so far?
  8. Aug 30, 2010 #7


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    I can see no good reason to use "Laplace transform". The problem is close to trivial with regular methods (find the general solution to the associated homogeneous equation and add a particular solution found by "undetermined coefficients".
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