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Math Help - Differential Equations

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    Member classicstrings's Avatar
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    Differential Equations

    A model for the number W of wombats on a island, t years after a initial 200 are settled there, takes into account the availability of wombat food. Under this model, dW/dt = (m - n - kW)W where m and n are related to the birth rate and death rate respectively, and k is a positive constant related to the amount of food available on the island.

    Suppose m = 0.10, n = 0.06 and k = 0.00005 and W < 800. Find an expression for the number of wombats on the island after t years. Find a general solution for the differential equation that represents the model for W > ((m-n)/k).

    What happens as t approaches infinity? Use calculus to find at what time, t, is the wombat population increasing most rapidly.

    ...

    I've tried to antidiff, by flipping to get dt/dw and then t = (1/(m-n-kw))logeW + c. I have to find a C value to find a general solution, and I get (t=0, w=200), c = (loge200 - 0.01)/(-0.04). Then I get stuck on simplifying to get a general solution.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by classicstrings View Post
    A model for the number W of wombats on a island, t years after a initial 200 are settled there, takes into account the availability of wombat food. Under this model, dW/dt = (m - n - kW)W where m and n are related to the birth rate and death rate respectively, and k is a positive constant related to the amount of food available on the island.

    Suppose m = 0.10, n = 0.06 and k = 0.00005 and W < 800. Find an expression for the number of wombats on the island after t years. Find a general solution for the differential equation that represents the model for W > ((m-n)/k).

    What happens as t approaches infinity? Use calculus to find at what time, t, is the wombat population increasing most rapidly.

    ...

    I've tried to antidiff, by flipping to get dt/dw and then t = (1/(m-n-kw))logeW + c. I have to find a C value to find a general solution, and I get (t=0, w=200), c = (loge200 - 0.01)/(-0.04). Then I get stuck on simplifying to get a general solution.
    \frac{dW}{dt} + (n - m)W = -kW^2

    is a Bernoulli differential equation. (They are of the form: y ^{\prime} + P(x)y = Q(x)y^n where n is an integer.)

    Use the substitution:
    W(t) = \frac{1}{y(t)}

    Then
    \frac{dW}{dt} = -\frac{1}{y^2} \frac{dy}{dt}

    and
    -\frac{1}{y^2} \frac{dy}{dt} + \frac{(n - m)}{y} = \frac{-k^2}{y^2}

    or
    \frac{dy}{dt} - (n - m)y = k^2

    which I'm sure you can solve from here.

    -Dan
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