Math 385 Section C2 Test 2, Spring 2003

Wednesday 9 April, 55 minutes in class, worth 20%.
You may not use books, notes, calculators or computers on the test.

Study sessions in 143 Altgeld: Monday 7 and Tuesday 8 April, 7-8pm

Material
Sections 3.1-3.6, 3.8, and Supplement on Orthogonality.

What and how to study

First make summary notes of the important ideas and methods from each section.
Make a summary table of the main properties and results in Sections 3.4 and 3.6, on mechanical vibrations (there are four cases, since you can either have damping or not, and forcing or not; we always assume the forcing function is periodic). And you should definitely work through the derivations of the solution formulas, in these four cases. Also, what is resonance? what is practical resonance?
You should aim to feel confident solving any "reasonable" nonhomogeneous problem, using the method of undetermined coefficients or the method of variation of parameters (for which you should learn the formula for y_p).
In both methods, you first find y_c and then find y_p, then apply the initial conditions to evaluate the constants in y_c. Note: We know how to find y_c for two types of equation: constant coefficient, and equidimensional.

The text contains plenty of material that we did not cover in class, and you generally do not need to learn this material for the test. Conversely, we did cover some material in class that is not in the text, and you need to study this material. Here are some detailed guidelines:

• You can ignore Sec. 3.1 Th. 3 and Sec. 3.2 Th. 3, on Wronskians.
• In class we covered linear independence of exponential functions and sine and cosine, without using Wronskians (recall that instead we assumed the functions were linearly dependent and worked directly with the linear combination to obtain a contradiction; see questions in the "Section 3.2 Summary" at the handouts page).
• You should ignore Sec. 3.5 Rule 1 as it is stated in the textbook. It is much better to use Rule 1 as stated in the handout "Undetermined Coefficients Summary" on the handouts page.
• You should study the Supplement on Orthogonality of Eigenfunctions, which we covered after Section 3.8. You are not responsible for the proofs, but you should know how to apply the Orthogonality of Eigenfunctions Theorem, like in the given examples.

Understand the meaning of the theorems and definitions, because they give structure to the examples that we consider. (But generally, it is not worth remembering the exact statements of theorems.) I will not ask you to prove any theorems.

If you just work problems without first making summary notes, you will probably be wasting your time, because you won't have a mental framework into which to fit the examples you are working. It is the underlying methods and techniques that we care about, with the reason we work examples being to show we understand the techniques.