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# Use differential equations to determine population under given growth assumptions

The simplest population growth law is given by

where is the population and is a constant dependent on the type of population in question. A more involved growth law in which the population is subject to a maximum constraint , gives an equation for population growth of

where is a constant, or possibly a function of time.

Express the population as a function of in each of these growth laws (with and both constant). Prove that the result in the second growth law can be expressed as:

where is a constant and is the time at which .

Proof. For the first growth law,

This has solutions of the form

where at time .

For the second, more complicated, growth law we have

This is a Bernoulli equation (as seen in this exercise, Section 8.5 Exercise #13) so we know where is the unique solution to

Thus, we are looking for the unique solution of

Using Theorem 8.3 (page 310 of Apostol) for the solutions of first-order linear differential equations, we have

Therefore,

So, if we have , then and , so

where