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.
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
You would then follow the same method you learned previously for finding a tangent line .
You would then follow the same method you learned to find second derivatives when applying implicit differentiation.
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.
Ex: y is directly proportional to x , y is inversely proportional to x .
Proportional |
Inversely Proportional |
1) Base case $y=\mathit{kx}$ |
1) Base case $y=\frac{k}{x}$ |
2) Proportional to the square of $y=k{x}^{\textcolor[rgb]{}{2}}$ |
2) Inversely proportional as the square of $y=\frac{k}{{x}^{\textcolor[rgb]{}{2}}}$ |
3) Proportional to the square root of $y=k\sqrt{\textcolor[rgb]{}{x}}$ |
3) Inversely proportional as the square root of $y=\frac{k}{\sqrt{\textcolor[rgb]{}{x}}}$ |
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.