Math 111.01/02 (Applied Mathematical Analysis I)

Fall 1998

Plan for Week 2

We start this week with a topic that should be familiar from your single-variable calculus course -- the idea of a separable differential equation, which can be "solved" by separating the variables, integrating both sides, and solving the resulting equation for y as a function of t.

This procedure often fails to produce a formula for y(t) because any one of the three steps may not be possible. First, many first-order equations are not separable. Second, even if the equation is separable, it may not be possible to carry out one or both integrations in closed form. Third, even if both integrals can be evaluated symbolically, it may not be algebraically possible to solve the resulting equation for y.

The rest of this week we will see that it is (almost) always possible to find a numerical or graphical solution to a first-order initial value problem. Thus, we can always get a pretty good idea of the nature and appearance of solution functions, even when we can't get a symbolic formula. Furthermore, numerical solutions are useful even if we subsequently find a formula: First, it is almost always easier to find a numerical solution, and second, what we learn from the numerical solution serves as a check on subsequent algebraic calculation.

The basic idea behind numerical solutions is exactly the same as the idea behind the direction field: At every point of the (t,y)-plane, we know the slope of the solution passing through that point. If we use that information at an initial point, we can proceed a short distance in that direction and find a new point (approximately) on the solution curve. That becomes a new "initial" point, and we simply start over to get the next point, and the one after that, and so on. We will also explore and use some more sophisticated methods that involve better estimates of the actual slope from each point to another point on the curve, as opposed to a nearby point along the tangent line.

Here is the syllabus for Week 2:

Week 2 Date Topic Reading Activity
M 9/7 Separable DEs 2.3
W 9/9 Numerical solutions 8.1a, 8.3a,
8.4a
F 9/11 Numerical solutions Lab: Numerical Solutions
of Initial Value Problems

Notes

  1. Your next homework papers will be turned in on Monday, September 14. Those papers should include solutions to all problems in the assignment below. The assignment dates are start dates.
  2. Your textbook has far too many answers in the back of the book. If you look there first, you will subvert the learning process. If you know your answers are correct, you will never need to look there. In general, no homework problem will be given full credit unless you have written an explanation of why you know it is correct. (Exceptions to this rule are the exercises whose numbers appear in parentheses.) For example, an acceptable explanation for a solution of a differential equation is that you have substituted the proposed solution into the original equation and found that it satisfies the equation -- but you have to show your work. Two examples of unacceptable explanations: (a) It matches the answer in the back of the book. (b) I did the work again and it came out the same.
  3. Submit your Week 2 lab report (the Maple file) via e-mail by the end of the day Wednesday, September 16. This will be your first graded lab activity.
  4. Remember to submit your first journal entry by e-mail on Friday, September 11.

Assignments

David A. Smith <das@math.duke.edu>

Last modified: September 14, 1998