   Chapter 13.5, Problem 42E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Spread of disease An isolated community of 1000 people susceptible to a certain disease is exposed when one member returns carrying the disease. If x represents the number infected with the disease at time t (in days), then the rate of change of x is proportional to the product of the number infected, x, and the number still susceptible, 1000 — x. That is, d x d t = k x ( 1000 − x ) or d x x ( 1000 − x ) = k   d t (a) If k = 0.001, integrate both sides to solve this differential equation.(b) Find how long it will be before half the population of the community is affected.(c) Find the rate of new cases, dx/dt, every other day for the first 13 days.

(a)

To determine

To calculate: The solution of the differential equation dxx(1000x)=kdt if k=0.001. When the specific community of 1000 people those are suffering from several disease is exposed when one of the member returns carrying the disease.

Where, x represents the number of people infected by the disease at any time t (in days).

Explanation

Given Information:

The provided differential equation is,

dxx(1000x)=kdt

Formula used:

A separable equation is of the form:

g(y)dy=f(x)dx

Integration both sides yield the solution of this equation.

According to the formula 13 of the table,

duu(au+b)dx=1bln|uau+b|+C

Calculation:

Consider the provided differential equation,

dxx(1000x)=kdt

The equation is in separable form. So, integrate both sides to get,

dxx(1000x)=kdt

Now, use formula 13 on the left side as,

11000ln|x1000x|=0.001t+C1ln|x1000x|=t+C2x1000x=eteC2x1000x

(b)

To determine

To calculate: The time before half the population of the community is affected Where the specific community of 1000 people those are suffering from several disease is exposed when one of the member returns carrying the disease.

(c)

To determine

To calculate: The rate of new cases dxdt for every alternate days till first 13 days Where the specific community of 1000 people those are suffering from several disease is exposed when one of the member returns carrying the disease and differential equation is given by dxx(1000x)=kdt if k=0.001.

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