Solve the Lorenz equations given in Section 5.12.1 for r =1, using ode23.  Use the Adams–Bashforth–Moulton method to solve dy/dt=−5y, with y=50 when t =0, in the range t =0 to 6. Try step sizes, h, of 0.1, 0.2, 0.25, and 0.4. Plot the error against t for each case. What can you deduce from these results with regard to the stability of the method? The exact answer is y=50e−5t. The following first-order differential equation represents the growth in a population in an environment that can support a maximum population of K:

D N / d t = r N (1−N/K)

where N(t)is the population at time t and r is a constant. Given N =100 when t =0, we use the MATLAB function ode23, solve this differential equation in the range 0 to 200, and plot a graph of N against time. Take K =10,000 and r =0.1.

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