Math Homework #1 Solution .

$30.00

Description

  • Part I (50%)

This part is required to be submitted in class.

  1. Exercise 5.1.1.a

  1. Exercise 5.1.3:b,d

  1. Exercise 5.1.6

  1. Exercise 5.1.7

  • Part II (50%)

Population growth is described by an ODE of the form y0(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

y0

y

t 50;

(1)

(t) = r(1 K )y; 0

<

  • y(0) = y0

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

y0K

y(t) = 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

  • = 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.