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Math Help - Systems of ODEs

  1. #1
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    Systems of ODEs

    Using the Laplace transform, solve the initial value problem:
    y1 = -6y1 + 4 y2 (1)
    y2 = -4y1 + 4 y2 (2)
    y1(0) = -2
    y2(0) = -7

    Solving:
    (1) becomes: sY1 y1(0) = -6Y1 + 4Y2 => sY1 +2 = -6Y1 + 4Y2 (3)
    (2) becomes: sY2 y2(0) = -4Y1 + 4Y2 => sY2 + 7 = -4Y1 + 4Y2 (4)

    Two equations with 2 unknowns, so starting with (4), we get:
    Y1 = [7-(4-s)Y2]/-4

    Plugging into (3), we get:
    Y2 = [7s+50]/[(s+4)(s-2)]

    At this point, I can use partial fractions to determine A/(s+4) and B/(s-2), but I cant help but think I messed up the math somewhere as I cant match the stated answer of:
    y1(t) = 2e**-4t 4e**2t
    y2(t) = e**-4t 8e**2t
    The (s+4) and (s-2) terms of my answer makes me think Im tantalizingly close, but any help on where I messed up will be much appreciated.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by dsprice View Post
    Using the Laplace transform, solve the initial value problem:
    y1’ = -6y1 + 4 y2 (1)
    y2’ = -4y1 + 4 y2 (2)
    y1(0) = -2
    y2(0) = -7

    Solving:
    (1) becomes: sY1 –y1(0) = -6Y1 + 4Y2 => sY1 +2 = -6Y1 + 4Y2 (3)
    (2) becomes: sY2 – y2(0) = -4Y1 + 4Y2 => sY2 + 7 = -4Y1 + 4Y2 (4)

    Two equations with 2 unknowns, so starting with (4), we get:
    Y1 = [7-(4-s)Y2]/-4

    Plugging into (3), we get:
    Y2 = [7s+50]/[(s+4)(s-2)]

    At this point, I can use partial fractions to determine A/(s+4) and B/(s-2), but I can’t help but think I messed up the math somewhere as I can’t match the stated answer of:
    y1(t) = 2e**-4t – 4e**2t
    y2(t) = e**-4t – 8e**2t
    The (s+4) and (s-2) terms of my answer makes me think I’m tantalizingly close, but any help on where I messed up will be much appreciated.
    The error is in:

    Plugging into (3), we get:
    Y2 = [7s+50]/[(s+4)(s-2)]


    this is wrong. It should be:

    Y_2(s) = -\;\frac{7s+34}{(s+4)(s-2)}

    Which I believe gives:

    y_2(t)=e^{-4t}-8e^{2t}

    CB
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  3. #3
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    Systems of ODEs

    Drats! And I even checked my math at least 10 times! Thanks. But now when I try to solve for Y1, I get:

    Y1 = [-2s 18]/[(s+4)(s-2)]

    This doesnt seem quite right. And Ive once again checked my math at least 10 times and am running out of paper. J
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by dsprice View Post
    Drats! And I even checked my math at least 10 times! Thanks. But now when I try to solve for Y1, I get:

    Y1 = [-2s – 18]/[(s+4)(s-2)]

    This doesn’t seem quite right. And I’ve once again checked my math at least 10 times and am running out of paper. J
    it is:

    Y_1(s)=-\;\frac{2s+20}{(s+4)(s-2)}

    CB
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  5. #5
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    Thanks again. It still took me several attempts to finally get the same result. I seem to be making math more difficult than it needs to be!
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