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## Homework Statement

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A carrier of an infectious disease joins a herd of 500 initially uninfected cattle. At any

instant in time, the rate at which the disease spreads dx/dt is known to be proportional to

the product of:

(i) the number of infected cattle x(t); and

(ii) the number of uninfected cattle.

If the number of cattle infected after 4 days is 50, how many will have been infected after

6 days?

## Homework Equations

## The Attempt at a Solution

I interpreted the problem as the following D.E:

dx/dt=k*x*(500-x); where k is a constant to be determined

And the initial condition: x(0)=0

And the given information: x(4)=50

The equation looks to be separable:

dx/(500*k*x-k*x^(2)) = dt

but I cannot integrate the left hand side.

Is my interpreted D.E correct? If yes how to integrate the left hand side?

Appreciate your help.

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