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Differential equations-system of equations, cleaning up the great lakes...

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Mar24-08, 09:57 PM
P: 11
1. The problem statement, all variables and given/known data

The idea of the problem is to find out how long it would take to flush the Great Lakes of pollution. They're set up as a series of five tanks and you are given inflow rates of clean water, inflow rates from the other tanks, and outflow rates. See attached figure.

2. Relevant equations

dA/dt=ratein-rate out where A is the amount of pollution at time t

3. The attempt at a solution

first I wrote equations for each lake. rather than using A as my variable, I used the first letter of each lake (with n for Ontario) to stand for the amount of pollution in the given lake at time t. this gives...


dm/dt= -38m/1180

dh/dt=15s/2900 + 38m/1180 - 68h/850

de/dt=68h/850 - 85e/116

dn/dt=85e/116 - 99n/393

rearranging and pulling out the differential operator leads to the following system

(D + 38/1180)[m]=0
(D - 68/850)[h] + 38m/1180 - 15s/2900=0
(D + 15/2900)[s]=0
(D + 85/116)[e] -68h/850=0
(D + 99/393)[n] - 85e/116=0

and there you have it. A system of five equations with five variables. It seems like it should be fairly straightforward to solve from here, but I can't figure out what to do next. I'm stuck!!
Attached Thumbnails
great lakes.JPG  
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Mar24-08, 10:50 PM
P: 11

here's the image that shows the great lakes and their flow rates so you don't have to view the attachment

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