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Using Stepfunction to solve IVP

  1. Oct 25, 2009 #1
    1. The problem statement, all variables and given/known data

    I am given the task for finding the laplace transformation given

    [tex]y'' + 4y = 8t^2 [/tex] if 0<t<5 and 0 if t>5 y(1) = 1+cos(2) [tex]y'(1)= 4-2sin(2)[/tex]

    2. Relevant equations

    3. The attempt at a solution

    I know that the above problem can be written [tex]y''+4y=8 \cdot u(t)[/tex]

    where u(t) is the step function.

    But what is my next step?

    [tex]\mathcal{L}(y''+4y) = \mathcal{L}(8t^2)[/tex]

    which equals

    [tex]\mathcal{L}(y'') + \mathcal{L}(4y) = 8\mathcal{L}(t^2)[/tex]

    [tex]\mathcal{L}(y'') = s \cdot \mathcal{L}(y') - y'(1) [/tex]

    [tex]\mathcal{L}(y') = s \cdot \mathcal{L}(y) - y(1) [/tex]

    not sure what to do about the left side by the rightside that is [tex]Y(s^2+ 4) = \frac{16}{s^3} + s(1+cos(2)) + (4-2sin(2)) [/tex]

    then [tex]Q(s)= \frac{1}{s^2+4}[/tex]

    then [tex]Y(s) = s(1+cos(2)) + 4-2sin(2)) \cdot \frac{1}{s^2+4} + \frac{16}{s^3} \cdot \frac{1}{s^2+4}[/tex] Is this true or am I totally fubar?
     
    Last edited: Oct 25, 2009
  2. jcsd
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