Introductory Programming
Fall 2005

Today's instructions

1) create a file named f.m in the usual place you put scripts

2) type in the following code

function dydt = f(t, y)
dydt = t + y;
end

3) evaluate this function from the command line and make sure
   it does what you expect.  

4) type 'help ode45' and read the documentation, or at least
   as much of it as you can stand

5) run 'ode45(@f, [0, 1], 0)'  What does this command do?

6) Alternatively, you can run '[T, Y] = ode45(@f,[0, 1], 0);'
   and then 'plot(T, Y)'

7) Use ode45 to find a numerical solution to the differential
   equation

   dx/dt = g (T - x) + d(T^4 - x^4) + el u^2

   with T = 293.15, g = 50, d = 2e-7, el = 800 and u=21.

   The initial condition is x(0) = 25.

   (See the problem statement on the next page.)

8) Here is the program I wrote to compute mouse populations:

a = 0.9;
B = [70 36 11 1 4 13 28 43 56] * 1e-4;
t0 = 0;
tend = 8;    % months
n = 1000
dt = (tend - t0) / n;

t = zeros(1, n+1);   % time
N = zeros(1, n+1);   % mouse population

t(1) = t0;
N(1) = 100;    

for i=2:n+1
    t(i) = t(i-1) + dt;
    index = floor(t(i-1)+1);
    b = B(index);
    dN = dt * (a * N(i-1) - b * N(i-1) ^ 1.7);
    N(i) = N(i-1) + dN;
end

Write a version of f.m that computes dN/dt for given values
of N and t, and use it to estimate the population of mice
over t = [0, 8] with the initial value N = 100.

Does the solution agree with what we got when we did the
integration ourselves?