1550078481-Ordinary_Differential_Equations__Roberts_

142 Ordinary Differential Equations

Recall that an epidemic ends at time t when I ( t) = 0. In this case,

when the epidemic ends the entire population is divided into two mu-

tually exclusive sets- the infected and the inoculated. Use SOLVEIVP

or your computer software to compute and graph the solution of the

differential equation (9) on the interval [O, 10], if J(O) = 1, /3 = .002,

N = 500, the inoculation function i(t) = at (this function indicates

that health personnel are able to inoculate a units of people per unit of

time), and (a) a = 30 and (b) a = 50. In each case, what is the duration

of the epidemic, t*, and what is the maximum number of infectives at

that time, I(t*)?

2. Suppose that the inoculation procedure (see exercise 1) accelerates
   over time so that i(t) = t + 30. Then the differential equation for the
   number of infectives becomes
   (10) I'(t) = f3[N - (t + 30)t - I(t)JI(t).
   Compute and graph numerical solutions of the differential equation (10)

on the interval [O, 10], if J(O) = 1, f3 = .002, and N = 500. What is the

duration of the epidemic, t*, and the maximum number of infectives at

that time, I(t*)?

3.5 Falling Bodies

On April 28, 1686, Isaac Newton presented and dedicated to the Royal Soci-
ety the first volume of a three volume work which he chose to title Philosophiea
Naturalis Principia Mathematica (Mathematical Principles of Natural Philos-
ophy). This work is almost always referred to simply as the Principia. The
first volume deals mainly with various problems of motion under the idealized
conditions of no friction and no resistance. In Book I of the Principia, Newton
formulated his three laws of motion and in Book III he expanded upon them.

NEWTON'S LAWS OF MOTION

Newton's three laws of motion are as follows:

First Law of Motion: "A body at rest tends to remain at rest,

while a body in motion tends to remain in motion in a straight line

with constant velocity unless acted upon by an external force."

Second Law of Motion: "The rate of change of momentum of

a body is proportional to the force acting on the body and is in the

direction of the force."

Third Law of Motion: "To every action there is an equal and

opposite reaction."