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

1. Homework Statement

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. Homework 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...

ds/dt=-15s/2900

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)=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!!

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