(Differential equation) Finding an exponential equation

  • #1

Homework Statement



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

Homework Equations





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:

Answers and Replies

  • #2
haruspex
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All looks right to me.
 

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