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Solution of Initial Value Problems, Laplace Transform

  1. Apr 29, 2005 #1
    Use the Laplace transform to solve the given initial value problem.

    y"+[w^(2)]y=cos2t, w^(2) does not equal 4; y(0)=1, y'(0)=0

    I tried doing the problem, and I got up to Y(s)=[(s^(3)+5s]/[s^(2)+w^(2)],
    which hopefully is correct. Now I'm having trouble using the Laplace
    transforms to finish solving the problem.

    Thanks for your help!
     
  2. jcsd
  3. Apr 30, 2005 #2

    dextercioby

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    What's the integral u have to solve to revert the Laplace transform...?

    Daniel.
     
  4. May 3, 2005 #3

    saltydog

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    Taking the Laplace transform of both sides:

    [tex]\mathcal{L}\{y^{''}+\omega^2 y=Cos[2t]\}[/tex]

    yields:

    [tex]s^2f-s+\omega^2f=\frac{s}{s^2+4}[/tex]

    with:

    [tex]\mathcal{L}\{y(x)\}=f(s)[/tex]

    Solving for f(s) yields:

    [tex]f(s)=\frac{s^3+5s}{(s^2+4)(s^2+\omega^2)}[/tex]

    Now, you can reduce this using partial fractions with quadratic denominators:

    [tex]\frac{s^3+5s}{(s^2+4)(s^2+\omega^2)}=\frac{As+B}{(s^2+4)}+\frac{Cs+D}{(s^2+\omega^2)}[/tex]

    When solving for A,B,C,and D, you'll come up against a homogeneous system for B and D, and the constraint imposed on [itex]\omega^2[/itex] will force you to make a conclusion about what B and D can be. Try solving it to completion and report the results here. If you don't, I'll wrap it up tomorrow.
     
  5. May 4, 2005 #4

    saltydog

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    Just some wrap-up:

    Solving for A and C:

    [tex]A=\frac{1}{\omega^2-4}[/tex]

    [tex]C=\frac{5-\omega^2}{4-\omega^2}[/tex]

    The expressions for B and D are:

    [tex]B+D=0[/tex]

    [tex]B\omega^2+4D=0[/tex]

    You know, I'm not sure about the part below deciding about B and D: Really, I think B and D HAVE to be zero in this case. Someone can correct me if my rational is not so.

    Noting that a homogeneous equation has a non-zero solution iff determinant=0, we get:

    [tex]4-\omega^2=0[/tex]

    However, since constraint above restricts such, only the trival solution is allowed. Thus:

    [tex]f(s)=A\frac{s}{s^2+4}+C\frac{s}{s^2+\omega^2}[/tex]

    Finally, taking the Laplace transform of both sides:

    [tex]y(t)=ACos[2t]+C(Cos[\omega t])[/tex]

    A plot is attached.
     

    Attached Files:

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