North American lake system. Consider the American system of two lakes: Lake Erie feeding into Lake Ontario. What is of interest is how the pollution concentrations change in the lakes over time. You may assume the volume in each lake to remain constant and that Lake Erie is the only source of pollution for Lake Ontario.

(a) Write down a differential equation describing the concentration of pollution in each of the two lakes, using the variables V for volume, F for flow, c(t) for concentration at time t and subscripts 1 for Lake Erie and 2 for Lake Ontario.

(b) Suppose that only unpolluted water flows into Lake Erie. How does this change the model proposed?

(c) Solve the system of equations to get expressions for the pollution concentrations, c1(t) and c2(t).

(d) Set T1 = V1/F1 and T2 = V2/F2, and then T1 = kT2 for some constant k as V and F are constants in the model. Substitute this into the equation describing pollution levels in Lake Ontario to eliminate T1. Then show that, with the initial conditions c1,0 and c2,0, the
solution to the differential equation for Lake Ontario is

c2(t) =k/k − 1c1,0(e^−t/(kT2) − e^−(t/T2)) + c2,0et/T2 .