Find the solution of the seasonal growth model.

Discussion in 'Differential Equations' started by fgvand94, Jun 6, 2022.

  1. fgvand94

    fgvand94

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    It's given that a model for seasonal growth is dp/dt = kPcos(rt - m) and asks to solve for the initial condition P(0)=P(subscript0). I wasn't sure how to make a subscript so to make it easier lets just say P(0) = P*. I'm not quite getting the answer that's in the book and I was wondering if someone could show me the steps so I can see what I'm doing wrong. The answer I got is almost what's in the book so I'm sure it's something small I'm overlooking/forgetting. This is what I got.

    dp/kP = cos(rt - m)dt

    Integrating both sides gives

    ln|kP|/k = rsin(rt - m) + C

    Multiplying out the k, putting both sides as a exponential of e and dividing by the k inside of the logarithm gives

    P = e^krsin(rt - m)e^kC/k

    if we let e^kC/k = A we get

    P = Ae^krsin(rt - m)/k

    Solving for the initial condition P(0) = P* gives

    A = P*k/e^krsin(-m)

    Plugging that into P gives

    P = P*e^krsin(rt -m) - krsin(-m) or P = P*e^kr[sin(rt - m) + sin(m)]

    The only difference is the book has k/r in the exponent on e not kr and I'm not sure why.
     
    Last edited: Jun 6, 2022
    fgvand94, Jun 6, 2022
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  2. fgvand94

    MathLover1

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    MathLover1, Jun 6, 2022
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    nycmathguy and fgvand94 like this.
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  3. fgvand94

    fgvand94

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    Oh lol. The r goes on the bottom cause after differentiation using the chain rule the r has to disappear to get back to the original expression meaning there has to be an r on the bottom. for some reason I just put it out front. I was mixing myself up thanks.
     
    fgvand94, Jun 6, 2022
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    MathLover1 likes this.
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