Method: Differential Equation Applications

Differential equation word problems do not have a one size fits all approach to the method of solving. Depending on the problem they could have you do any number of actions using the differential equation. Remember that a differential equation is just a fancy way of telling you that you are being given the derivative equation to work with. Most of what they will ask out of you will relate back to topics you have covered previously. Here is a list of some items to be on the lookout for.

• They could ask you how that rate of change is behaving at specific given moments. In those situations, you are usually plugging a given value into your differential equation to find the answer. Sometimes on an AP Calc exam, especially in the free response area, it is hard to trust that all you need to do is plug in a value, but sometimes it is really that easy.

It is important in this type of problem to identify what the two variables you are working with represent. I often will write the letters of my two variables down and then an equal sign with what that variable represents.

Ex:  B = weight and t = time

• They could ask you to find a tangent line at a given point.

You would then follow the same method you learned previously for finding a tangent line .

• They could ask you to find the second derivative. Often, they will use the differential notation for a second derivative when they do, (i.e., find $\frac{d^{2}B}{d{t}^2}$)

You would then follow the same method you learned to find second derivatives when applying implicit differentiation.

• They could ask you to find the “particular solution” to the given differential equation.

This is really asking you to perform the same method you learned for initial value problems. It may require you to apply the methods associated with separable differential equations we will learn in the next section. However, as you will find out in the next session, there is not much different between the method for finding the “particular solution” to a separable differential equation, and the initial value problem method. It is just a fancy way of telling you to find the +C value.

• You will also see them start to use some language from algebra that you have not seen in a while. When describing the differential equation, you will sometimes see them use the terms “proportional”, “directly proportional”, or “varies directly” and “inversely proportional”, “indirectly proportional” or “varies indirectly”. This language is a cue to tell you how the equation needs to be setup. There are two main families of setups, the proportional and the inversely proportional . Within those two families you can see some additional language following that main language. That additional language is telling you what to do to the independent variable ( x -variable) always. Also, when you read the sentence the dependent variable ( y -variable) is always before the proportional or inversely proportional language, and the independent variable ( x -variable) always come immediately after the language.

Ex: y is directly proportional to x , y is inversely proportional to x .

Most often on the AP Calc exam I have seen them use this language, and then give you the actual equation already setup; having you not need to do any of the setup. I have found this to be a little confusing, but that is what they do. Which means you need to be on the look out for it, and not let it trip you up.

I have seen this language very often used in the actual AP Calc class by teachers. In those situations, I have found the teachers do not give you the equation already setup, and they expect you to know how to set it up yourself.