1550078481-Ordinary_Differential_Equations__Roberts_

404 Ordinary Differential Equations

Since the amount of pollutant in each lake, qi (t), equals the concentration

of pollutant, ci(t), times the volume, 1/i, we h ave for each lake qi(t) = Ci(t)V'.i.

Using Figure 9.11 to write a n equation for the rate of ch ange of the amount

of pollutant in each la ke, we find

for Lake Superior

dqs = d(cs Vs) = RsCs - rsCs

dt dt

for Lake Michigan

dqm d( cm Vm)

dt dt

for Lake Huron

dqh d(chVh)

• = = RhCh + rsCs + rmem - rheh
  dt dt
  for Lake Erie
  dqe d(ee Ve )
  dt = dt = ReCe + rheh - reee
  for Lake Ontario
  dqa d(eaVa)
  dt = dt = RaCa + reee - raCa.
  Dividing each of these equations by the volume of the corresponding lake, we
  obtain the following system of differential equations for the concentration of
  pollution in the Great La kes

des R sCs - rses

dt Vs

• dem - RmCm-rmem
  dt Vm

(19)

deh RhCh + rses + r mem - rheh

dt vh

dee R eCe + rheh - reee

• dt Ve

dea RaCa + reee - raea

dt Va

Exercise 15. a. Write system (19) using matrix-vector notation. (The answer
will have the form c' =Ac+ b .)

b. Assume all pollution of the Great Lakes ceases-that is, assume Cs =

Cm = Ch = C e = Ca = 0. Use EIGEN or your computer software to find the

general solution of c' = Ac where the constant entries of A a re calculated

using the informatio n given above for the Great Lakes.