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You may do these assignments with any software. See instruction page for doing them on netmath.

1.  On the inclass exercises on exponentials (September 12), in problem 5 you are asked about solutions to the equation dN/dt = -2N + 4. Here we are asking you to graph solutions to a similiar equation dN/dt = -.5N + 6.3. Find the direction field plots for this equation, and graph the solution with N(0) = 2. Use a symbolic program to give the general formula for the solution.

2. On page 23, Taubes' suggests the model

dp/dt = ap(1-p)(p-1/2) = f(p).

Plot the direction fields for a = 5. Plot three solutions a) p(0) = .2,  b) p(0) = .6 and c) p(0) = 1.2. Now use a symbolic manipulator to solve the equation algebraically.(you should have one constant in the algebraic answer). Notice how hard it is to see the relationship between the algebraic answer and the behavior as t--> infinity.

3. (This is problem 1 on page 56). Let dp/dt = p(1-p). Draw the direction field, and plot solutions with a) p(0) = -1, b) p(0) = .5 c) p(0) = 2.

4. (This is problem 3 on page 57) Let dp/dt = e^p - 1. Draw the direction field and plot the solutions with a) p(0) = -1 p(0) = 1 and p(0) = 4. 5. Reading 2.1 had two complicated systems in it. Consider the first system (how T cells get infected by the virus):

dT*/dt = kVT - dT*
dV/dt = NdT* - cV.

The constants are difficult to determine from the article, because we could rescale things quite a bit and get the same behavior. Hence they are not important. The constants you need to determine are kT, d, N and c. To have a steady state, NkT = c. Pick a reasonable c and d from the article, choose N = 1 and kT = c. Draw the direction fields using for this example. How do we handle the problem that typical numbers for V are (10)^5? Plot a few solutions curves. We will solve second system in the next computer assignment.