A cattle rancher discovers a viral disease spreading among his 4000 cattle. Suppose the rate at which the disease spreads is proportional to the product of the number of cattle that have the disease and the number that do not. Write a differential equation that describes this scenario.

Question

A cattle rancher discovers a viral disease spreading among his 4000 cattle.  Suppose the rate at which the disease spreads is proportional to the product of the number of cattle that have the disease and the number that do not.  Write a differential equation that describes this scenario.

zugzwang · Answer

Let's represent.  Suppose we let x be the number of cattle with the disease...

zugzwang · Answer

And let the time (in whatever units) be represented by t.
Then the rate at which the disease spreads is
$$\frac{dx}{dt}$$

zugzwang · Answer

Now, if a and b are proportional, it only means that there exists a constant c such that

a = bc
Can you finish your differential equation from here?