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Description

Part I (50%)
This part is required to be submitted in class.

Exercise 5.1.1.a

Exercise 5.1.3:b,d

Exercise 5.1.6

Exercise 5.1.7

Part II (50%)
Population growth is described by an ODE of the form y^{0}(t) = ry(t), where r is the growth rate. In a typical population, the growth rate is not a constant, but is density dependent. For example, as the population grows, there might be less food available, and as a result the growth rate decreases. We consider the following Logistic Equation:

8
_{y}0
y
t 50;
(1)
(t) = r(1 _{K} )y; 0
<

y(0) = y_{0}
where 0 < y_{0} < K. Then the exact solution is given by
y_{0}K
^{y(t) =} _{y}_{0} _{+ (K} _{y}_{0}_{)}_{e }rt ^{:}
Solve IVP (1) with y0 = 1000, r = 0:2, K = 4000 numerically using Euler’s method. Choose the step sizes

= 10; 1; 0:1, respectively.
a) Compare the solutions to the exact solution in plots of population vs. time. Compare the actual maximal error max jy(ti) wij with the error bound predicted in Theorem 5.9 (p.271).
i
b) Discuss the behavior of the solutions as a function of h. What happens for very large step size h?
Requirements Print your pdf le and submit it in the discussion section. Submit to CCLE the code, for example: a MATLAB (or C++, C, etc) function euler.m that implements ALGORITHM 5.1 (p.267), and a MATLAB script main.m that solves the IVP (1) and plots the approximated solutions versus the exact one.