If then our slope field will just be a graph of the value of .  This is also easier to pick out if we recall that exponential growth only has one equilibrium at .  Otherwise the graph of the function is constantly increasing above the x-axis and constantly decreasing below the y-axis.  So when , the solution for all values of  is .  When , for all values of , the solution is .  Following this trend we can make a table with all solutions and plot our solutions.  This will result in the slope field above.

Looking at this slope field, let’s start by looking along the x-axis.  There appears to be no solutions plotted here.  This means that the function is undefined at these points.  We know that the function is at equilibrium if the tangent line (slope) is equal to zero.  So if we look at our solutions, we see that the tangent lines are horizontal at  for any  value.  And so this is our equilibrium solution.

If we make a table based off of the slope field given, it will look something like this.

Looking at this table we see a trend; when  (or when  - the same thing) the tangent line is .  This means we are at equilibrium when .

We are able to find the equilibrium points to a differential equation numerically.  We use slope fields because often times we will encounter complicated differential equations and this is the easiest way to find the equilibrium points or look at trends.

There is an equilibrium point at  but this is because if your population is at zero organisms, then there is no potential for growth (assuming that we are not considering spontaneous generation).  We see the tangent lines also have a slope of zero, meaning the function is at equilibrium, at .  This means that our carrying capacity is .

Our equilibria will be where the tangent lines have a slope of .  If we make a table we see that the tangent lines will be  when .

We are actually able to connect these line segments or start at an initial tangent slope and graph the solution curve.  This allows us to see, if we have an initial condition, what the solution will be and what trend the curve will follow.

When we are trying to integrate a differential equation, sometimes we have to use a method called separation of variables.  This is because we need to integrate with respect to each variable but are unable to do so when they are on the same side.  When we are able to get each variable on different sides with their respective differential (i.e.  or ), then we can integrate each side with respect to each variable.

By using separation of variables we are able to integrate to solve the differential equation.

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.

                                   (divided by  and multiplied by )

                                 (just rearranging)

                     (Let )

                                         (  is just a constant so we rename it )