Answer:

And if we integrate both sides we got:

Where C is a constant., we can rewrite the expression like this:

If we square both sides we got:

If we use the initial condition we have that:

And we can solve for C like this:


And now we can find the derivate of the function and we got:

Using the condition
we got:


k= 1
And then the model is defined as:

And for t =12 months we have:

Step-by-step explanation:
For this case we cna use the proportional model given by:

Where k is a proportional constant, P the population and the represent the number of months
For this case we know the following initial condition
and 
we can rewrite the differential equation like this:

And if we integrate both sides we got:

Where C is a constant., we can rewrite the expression like this:

If we square both sides we got:

If we use the initial condition we have that:

And we can solve for C like this:


And now we can find the derivate of the function and we got:

Using the condition
we got:


k= 1
And then the model is defined as:

And for t =12 months we have:
