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Math Help - differential equation

  1. #1
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    differential equation

    A neglected large lawn contains a certain type of weed. The area of the lawn covered by the weed at time t years is Wm. The rate of increase of W is directly proportional to W.

    Write down a differential equation that is satisfied by W.

    OK now i know the answer is [dw/dt = KW] because these questions go by the same format all the time

    BUT

    I need to know WHY it is that. I don't want any nasty surprises when i go into my exam :/
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by djmccabie View Post
    A neglected large lawn contains a certain type of weed. The area of the lawn covered by the weed at time t years is Wm. The rate of increase of W is directly proportional to W.

    Write down a differential equation that is satisfied by W.

    OK now i know the answer is [dw/dt = KW] because these questions go by the same format all the time

    BUT

    I need to know WHY it is that. I don't want any nasty surprises when i go into my exam :/
    If one variable is directly proportional to another variable, it means that they are constant multiples of each other. So in this case, one variable is \frac{dW}{dt} and the other variable is W, so you have the equation \frac{dW}{dt}=kW, where k is a constant.

    Separate the variables and integrate: \int\frac{dW}{W} = \int k\,dt \implies \ln(W) = kt+C \implies W=e^{kt+C} \implies W=c_0e^{kt} (where c_0=e^C).
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