 UBC Calculus Online Course Notes 
A LoveHate RelationshipApplications of the Trigonometric FunctionsIn our investigation of the trigonometric functions so far, we saw that from the pattern of derivatives that the pair of functions satisfied the following connected relationships:
We will now investigate an example of a situation to which this example applies.
Let's suppose we are psychologists, and assign some rating to the emotional state of each of the lovers. The rating system might look something like this:
We also define the two lover's emotional states by denoting:
It is understood that both these variables can take on any value in the range {1,1). For example, if , Jack is quite interested in his partner, but not at the extreme of his passionate love for her. Now we turn to trying to model the relationship by using mathematical ideas. Since the two people are continually changing their minds, clearly, our description should involve changes, and rates of change. We might expect to make statements involving . Since the way that each person's feelings change depends on the current state of their partner, we would expect that the relationship would be linked. Indeed, we might anticipate that some differential equations are lurking in this example. To be more precise, we can make the following observations about:
This suggests that we might try out the relationship
We observe that the correct sign pattern is preserved by this relationship. A similar analysis can be done for Jill's side of the story. Indeed, you might want to fill in the entries in the table for Jill
For your consideration:Use the information provided to fill in the entries of the following table
This table should lead us to a relationship of the form
(Check to make sure that you understand the connection between the derivative of one variable and the value of the other variable.) Putting together the two equations, we have:
You will perhaps notice that these equations are familiar ! Indeed we started with these equations at the top of this page, and reminded you that they arose in our investigation of the trigonometric functions. Since we are already familiar with these equations, we know that a pair of functions that satisfy this set of coupled differential equations are
The ups and downs of the relationship is shown below. Jack is shown in blue, and Jill in pink.
It turns out, however, that a much wider variety of functions also satisfy the same linked relationship. Indeed, it can be shown that the functions below are also solutions to the same set of equations:
Here are any real constants, so that this is a much larger class of functions which include the choice we made above as a special case.. (It turns out that this class of functions represents the most general form of solutions to the above system of equations. We have not left out or forgotten any functions that might also "work". However, to understand why this is true, further work, beyond the scope of this first introduction is required.) We can see that these more elaborate functions are also solutions by taking derivatives and plugging into the differential equations. For example, we note that the derivative of the first function is:
This verifies that the first equation is satisfied. You are asked to verify that the second equation is also satisfied below. For your consideration:
Suppose that when Jack and Jill first met at , Jill was quite neutral but Jack instantly fell madly in love . We will use this initial condition to determine the values of the two constants . We simply plug in and set to get:
where we have used the fact that above. From these equations, it is easy to see that the constants should have values .
For your consideration:
