A Numerical Approach

For this approach we assume that students are working in a situation where they have access to and experience with general purpose hardware and software for finding numerical estimates to solutions of initial value problems. Ideally, they are working with a general purpose package -- for example, the TI-92, the TI-92 Plus, Maple, Mathematics, DERIVE, or MathCad. We use the TI-92 here. Click here for programs and help using the TI-92 to find numerical estimates for solutions of initial value problems.

We rewrite our system of equations as follows to make them easier to work with on the TI-92.

The TI-92 screen below shows how with the particular programs above the TI-92 can be set up to investigate this system of differential equations.

For a first experiment we tried

a = 1

b = 1

c = 1

x(0) = 0

y(0) = 1

and, after some experimentation, we worked with the time interval [0, 4]. The TI-92 screen below shows the result.

This give us an immediate partial answer to our question -- at least in some circumstances the water level on the left will rise above the water level on the right.

But notice that we still have lots of questions. The most important question is -- Why? We should also ask whether this always happens or we just happened to pick one particular situation in which it did happen.

[Four Approaches] [Graphical]