We must manipulate this differential equation to get each variable and its own side with its differential.  Once we do that we must integrate each side accordingly.

                                   (multiplied by  and multiplied by )

                     (Let )

                                  ( is just a constant so we will rename it )

We must manipulate this differential equation to get each variable and its own side with its differential.  Once we do that we must integrate each side accordingly.

                                                    (multiplied by  and )

                                            (expansion)

                                       (Let )

                                         (  is a constant so call it )

Just like integrating for a function of a single variable at a particular solution.  We are also able to solve using separation of variables and integrating for our particular solution.

We first must use separation of variables to solve the general equation, then we will be able to find the general solution.

                             (multiplied by )

                (Let )

Now we plug in our particular solution  to solve for our constant 

And so our solution is

                                 (multiplying by  and multiplying by )

                      (Let )

                                    (  is just a constant so rename it )

Now we plug in our point  to solve for .

So our solution at this point is:

We first must use separation of variables to solve the general equation, then we will be able to find the particular solution.

                                        (multiplying by  and )

                                  (Let )

Now we plug in our initial condition that we were given 

Now we will solve for  when 

Separable equations are what we call differential equations that we are able to solve by using separation of variables.

We will use separation of variables to derive the general solution for the exponential growth model.

                          Let 

                                      is just a constant so  will also just be some constant.  We let .

In the exponential growth model (in this case it would be called the exponential decay model), .  But the entire exponent can be negative;  causing an exponentially decreasing population until that population reaches zero.  In theory, it would continue into negative values but biologically we know this is not feasible.

The most simple exponential growth model only takes into account the population’s current state, , time, , and a growth/decay constant that is greater than zero .  It does not limit the population to a carrying capacity or take into account resource availability/ predator-prey interactions.