We begin by taking the derivative with respect to  of our function .

Next we will plug this value into our differential equation.

Next we will write all  in terms of .  Recall that our original function says that , we will plug this in for all  terms.

And so ,  is a solution to the differential equation .

Let us consider .  By taking the derivative we see that .  We will plug this back into the differential equation as well as subbing in .

And so  is a solution to the differential equation 

Not all differential equations will have solutions.  Many, if not all, first order differential equations will have solutions to them.  As we move into second and third order differential equations, however, some of these may have no solutions.

This is a second order differential equation.  Recall that differential equations are equations that have both a function and AT LEAST ONE of their derivatives.  Equations with more than one derivative, such as those with first, second, and third order derivatives, are still differential equations.  This second order differential equation includes the second order derivative.

Let’s consider the equation .  If we take the derivative we see that .  Let’s plug that in for  in our differential equation as well as substitute .

And so the function  is a solution to the differential equation .

Consider .  The derivative of this function is .  Now let’s plug this into our differential equation and also let’s replace  with .

And so our solution to the differential equation  is .

Of the above examples, this is the only one that has a rate of change over time.  A population’s growth will vary over time depending on several factors such as resources available, predator/prey interactions, and carrying capacity.  The exponential growth model is in fact a differential equation:

Where  is the growth rate and  is the current population.  This differential equation has the solution .  Often times (if not all) population’s cannot grow exponentially forever, and so we also have a differential equation that is the logistic growth model which takes into account carrying capacity:

Where  is the carrying capacity.  The solution to this differential equation is .

A slope field is a visual representation of a differential equation in two dimensions.  This shows us the rate of change at every point and we can also determine the curve that is formed at every single point.  So each individual point of a slope field (or vector field) tells us the slope of a function .

We use slope fields when the differential equation we are given is too complicated to solve.  By plotting solutions of differential equations, we can see trends of our function , we can find equilibrium points, carrying capacities, etc.

The way to go about this problem is to make an  table and plug in  and  into our differential equation to find solutions.  We then plot the solutions at these points in our table.  It is important to note that our solutions are slopes, so we draw small line segments or vectors with the slope from our solution at each point.

Here is a sample of what your table should look like:

If we continue on like this, we will see that the slope field given above is the corresponding slope field for this differential equation.