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Math Help - Solving a piecewise function by Laplace transform

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    Solving a piecewise function by Laplace transform

    Hello, I am having great difficulty with the following problem:

    y''+6y'+5y=g(t)
    where g(t)=t for 0<t<2 and g(t)=0 for t>2. Initial values are y(0)=1 and y'(0)=0.

    I wrote g(t) as t-t\cdot U(t-2) and used Laplace transforms to get

    (s^2+6s+5)\mathcal{L}\{y\}-s-6=\frac{1}{s}-e^{-2s} \left(\frac{1}{s^2}+\frac{2}{s} \right).

    Originally I distributed the e^{-2s}, solved for \mathcal{L}\{y\} and ended up with some crazy partial fractions, but I think this definitely can't be right as I'm supposed to end up with another piecewise function as my answer. Could anyone point me in the right direction? I have never done a piecewise function with functions of t as the values.
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    Re: Solving a piecewise function by Laplace transform

    Quote Originally Posted by Ragnarok View Post
    Hello, I am having great difficulty with the following problem:

    y''+6y'+5y=g(t)
    where g(t)=t for 0<t<2 and g(t)=0 for t>2. Initial values are y(0)=1 and y'(0)=0.

    I wrote g(t) as t-t\cdot U(t-2) and used Laplace transforms to get

    (s^2+6s+5)\mathcal{L}\{y\}-s-6=\frac{1}{s}-e^{-2s} \left(\frac{1}{s^2}+\frac{2}{s} \right).

    Originally I distributed the e^{-2s}, solved for \mathcal{L}\{y\} and ended up with some crazy partial fractions, but I think this definitely can't be right as I'm supposed to end up with another piecewise function as my answer. Could anyone point me in the right direction? I have never done a piecewise function with functions of t as the values.
    It might help you to try to solve the two following DEs...

    \displaystyle \begin{align*} y'' + 6y' + 5y = t \end{align*} and \displaystyle \begin{align*} y'' + 6y' + 5y = 0 \end{align*}.
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    Re: Solving a piecewise function by Laplace transform

    Okay, I get the solutions

    y=\frac{-6}{25}+\frac{1}{5}t-\frac{13}{50}e^{-5t}+\frac{3}{2}e^{-t}

    for the first one and

    y=\frac{5}{4}e^{-t}-\frac{1}{4}e^{-5t}

    for the second.

    These are similar in form to the solution I got in my original attempt at the problem, though I had crazy coefficients. I just don't understand what to do next as I think we're supposed to end up with a function of the form

    f(t-a)\mathclass{U}(t-a)

    and then rewrite it in piecewise form.
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    Re: Solving a piecewise function by Laplace transform

    I just solved this problem with the help of my teacher. Thanks!
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