|
|
|
 |
|
|
|
AP Calculus Featured Question: January 2005
|
|
| |
by Ben Klein
Davidson College
Davidson, North Carolina
|
|
| |
Solving Systems of Differential Equations: 1994 BC6
In essence, problem 6 on the 1994 AP Calculus BC Examination asked students to solve a system of differential equations. With the current AB Course Description, this kind of problem might appear on an AB examination, and so the problem is now appropriate for both AB and BC students. In fact, the same is true for the extensions of the original problem we consider here.
In this Featured Question, we assume that students know that the solution to the initial value problem
is 
. This differential equation, of course, describes exponential growth.
We begin this month's problem with a "preface" that provides a solution technique for a slight generalization of the initial value problem given in the paragraph above. Then we state the two parts of the original problem, provide an alternative solution to the original problem, and finally ask students to solve a pair of systems of differential equations that are similar to the one presented in the original problem. The solutions to one of these systems are similar to those in the original problem. The solutions to the other system are quite different.
Students who work on this Featured Question should know that there are general methods to solve systems of differential equations like those we consider here. These methods are typically introduced in a first course on ordinary differential equations and are very powerful. The methods rely on some concepts from linear algebra and so provide a nice connection between algebra and analysis.
We will not, in this Featured Question, reproduce the "official" solutions for the two parts of the original problem. The result contained in the following preface allows a much simpler solution. In particular, the result provides an alternative to using separation of variables.
Preface
Follow the given steps to show that if a is not zero, the solution to the initial value problem
is 
.
STEPS:
- Use the change of variable
to reduce the given initial value problem to an initial value problem that describes exponential growth.
- Solve the simpler initial value problem for u(x).
- Use the solution, u(x), from step (2) and the change of variable from step (1) to solve the original initial value problem.
BC6
Let f and g be functions that are differentiable for all real numbers x and that have the following properties:
(A) Prove that 
for all x.
(B) Find f(x) and g(x). Show your work. [In this part, students should first use part (A) and then use the result in the preface to solve the problem.]
Parts (A) and (B) constitute the original problem. Part (C) below provides an alternative solution for part (B). Parts (D), (E), and (F) introduce two more systems of differential equations. You will find that the observation made in part (C) obviously simplifies the problem, but the corresponding observation in part (F) is not so obvious.
(C) Show that 
and use this fact to find f(x) and g(x).
We now present two more systems of differential equations. The first of these shows that two systems of differential equations can look very similar but have very different solutions. As you will see, it is easy to solve this first system.
(D) Let f and g be functions that are differentiable for all real numbers x and that have the following properties:
Find f(x) and g(x).
(E) Let f and g be functions that are differentiable for all real numbers x and that have the following properties:
Find f(x) and g(x) by first finding 
, then finding and solving an initial value problem whose solution is f(x), and finally finding g(x).
(F) If f(x) and g(x) are the functions in part (E), show that 
. Then use this observation to find f(x) and g(x).
Note: As promised above, the simplifying observation in part (F) is not as obvious as that in part (C). In fact, it probably seems a little mysterious. The observation is actually motivated by a straightforward application of an important concept from linear algebra and is not really mysterious at all in that context.
Complete the question before viewing the answers and explanation!
Ben Klein is currently the Beverly F. Dolan Professor of Mathematics at Davidson College in Davidson, North Carolina, where he has taught since 1971. Ben's relationship with AP Calculus began in 1990 when he served as a Reader at Clemson University. He has attended every Reading since then and has served as a Table Leader in recent years. In 2003, he completed a four-year term on the AP Calculus Development Committee.
|
|
|
|
|
|
Course Home Pages