Learning Differential Equations
Differential equations is the fourth course in a sequence. It differs from
the three calculus courses that preceded it in that:
- More theory is used in the solutions of problems.
- More problems involve multiple steps.
- The solution algorithms are considerably more complex.
- Mistakes and omissions yield either impossible completions or trivial
ones that don't merit partial credit for work beyond the mistake.
As a consequence of these differences, one needs to change the focus of
one's study time away from computations. It may feel good to find
the missing minus sign in three pages of algebra, but the half hour that
you use to do so would have been better spent reviewing the major steps in
the solution and the ideas behind it. You will probably never again drop
the minus sign in exactly the same place, so finding it will not help you
solve future problems. However, the procedures will come up problem after
problem, so learning them well will pay off.
Should your quiz and exam grades be less than you hoped, you might try the
following procedures, which have worked for your predecessors in this
course.
- Make 3 by 5 cards to study from. Cards should indicate when a procedure
is used, as well as all the steps in it and the type of result you should
expect at each stage. As you follow that procedure, note on the cards things
that you tend to forget to do, so you won't omit them. Make cards with
definitions of new words, statements of theorems and explanations of how
to use the theorems. Review the cards at least every other day to see if
you still remember all the data.
- Adopt a managerial policy towards arithmetic and delegate it to Maple
after you have done a few problems by hand. Spend your time looking
at the big picture and supervising, rather than being a slave to the
numbers. Know the kind of answer that Maple should produce, and
check that the answer fits that pattern, rather than worrying about
exactly how you would do the computation by hand.