Homework #1: If y, and y, are solutions of ay + by + cy = 0, prove that C₁y₁ + C₂y₂ is also a

solution.

So back to the estuary problem.

We now can write the general solution to the estuary equation, but will need boundary conditions to get a

particular solution. First thing to notice is that W, the waste load, does not appear in the ODE. The small

element Ax, see on Figure 6-2 in the original analysis of the mass balance, was arbitrarily chosen outside

the zone of waste discharge. So we really have two separate estuary zones to analyze; i.e., upstream and

downstream of the plane of waste discharge.

Then we have, switching from C for waste concentration to L for waste BOD concentration and writing

the general solutions for equation 3 in the upstream and downstream regions:

L₁ = C₁e¹² +₂e;

L₁ = C₂e¹*+C₂e;

In which ₁

T

and

upstream region

call it region I

U

=

= 2/1 (¹ + √/¹

1

2E

r =

- 00 < x≤0

2 2E

0≤x < 00

1 +

= 2/1 (1 - √ 1 + AKE)

4KE

downstream region

call it region II

OK to write, Both solutions will hold at

as will be seen via choice of BC's.

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