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(Differential equation) Finding an exponential equation

  1. Feb 26, 2013 #1
    1. The problem statement, all variables and given/known data

    A 44 gallon barrel of oil develops a leak at the bottom. Let [itex]A(t)[/itex] be the amount of oil in the barrel at a given time [itex]t[/itex]. Suppose that the amount of oil is decreasing at a rate proportional to the product of the time elapsed and the amount of oil present in the barrel.

    a. Give the mathematical model for A
    b. Find the general solution of the differential equation
    c. Find the particular solution for the initial condition

    2. Relevant equations



    3. The attempt at a solution
    a.
    [itex]\frac{\partial{A}}{\partial{t}} = -Akt[/itex]​
    b.
    [itex]\int{\frac{1}{A}\,dA} = \int{-kt\,dt}[/itex]
    [itex]\ln{A}=\frac{kt^2}{2}+C[/itex]
    [itex]A(t)=Ce^\frac{-kt^2}{2}[/itex]​
    c.
    [itex]44=Ce^0[/itex]
    [itex]A(t)=44e^\frac{-kt^2}{2}[/itex]​

    Is this correct? I'm not sure if I came up with the right [itex]\frac{\partial{A}}{\partial{t}}[/itex].
     
    Last edited: Feb 26, 2013
  2. jcsd
  3. Feb 26, 2013 #2

    haruspex

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    All looks right to me.
     
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