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Math Help - Solving Systems Using Laplace Transforms

  1. #1
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    Solving Systems Using Laplace Transforms

    Please help

    Use the Laplace Transforms to solve the initial value problem

    x'= x+ 2y, y'= x+ e^(-t); x(0)= 0, y(0)= 0
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  2. #2
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    Quote Originally Posted by googoogaga View Post
    Please help

    Use the Laplace Transforms to solve the initial value problem

    x'= x+ 2y, y'= x+ e^(-t); x(0)= 0, y(0)= 0
    Let \int_0^{\infty}e^{-st}xdt = F(s) \mbox{ and }\int_0^{\infty} e^{-st}ydt = G(s).

    Then,
    \int_0^{\infty} e^{-st}x' dt = \int_0^{\infty} e^{-st}(x+2y)dt
    \int_0^{\infty} e^{-st}y' dt = \int_0^{\infty} e^{-st}(x+e^{-t})dt

    Thus,
    -x(0)+sF(s) = F(s)+2G(s)
    -y(0)+sG(s) = F(s)+\frac{1}{s+1}
    Thus,
    F(s)(s-1)=2G(s)
    Substitute,
    sG(s) = \frac{2G(s)}{s-1}+\frac{1}{s+1}

    You finish it.
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