## Description

## 1 Part I (50%)

This part is required to be submitted in class.

(1) Exercise 5.1.1.a

(2) Exercise 5.1.3:b,d

(3) Exercise 5.1.6

(4) Exercise 5.1.7

## 2 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:

y

0

(t) = r(1 −

y

K

)y, 0 ≤ t ≤ 50;

y(0) = y0

(1)

where 0 < y0 < K. Then the exact solution is given by

y(t) = y0K

y0 + (K − y0)e−rt .

Solve IVP (1) with y0 = 1000, r = 0.2, K = 4000 numerically using Euler’s method. Choose the step sizes

h = 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

i

|y(ti) − wi

| with the error bound predicted in Theorem 5.9 (p.271).

b) Discuss the behavior of the solutions as a function of h. What happens for very large step size h?

Requirements Print your pdf file 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.

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