This gives more details about the final exam: It's a bit more than a review agenda, because it wants to make your review a deepened learning experience. So it includes hints about HOW you can learn and review wisely. Material covered consists basically of the union of the material listed for the individual exams (plus one small item of new stuff): From chapter 1 you should know about initial value problems, what the order of an ODE is and that usually you need n initial conditions for an n-th order ODE such as to determine the n constants for the general solution and get exactly one solution to the IVP. (Ch. 1 does this for 1st order only, notice the more specific existence and uniqueness theorem given there. But you have seen the principle for higher order linear ODEs as well.) You should understand direction fields and know how the Euler method works (e.g. calculating a single Euler step, or drawing/matching a direction field for/with an ODE) From chapter 2 you need to be able to see, given a 1st order ODE, whether it is separable / linear / homogeneous / Bernoulli / or whether the v=ax+by substitution applies. You also should know when the methods you learned do NOT apply to a particular ODE. (For instance I might give you 3 ODE's and ask you to solve *one* of them, at your discretion. The list may then contain one or two that you actually cannot solve at all.) And of course you need to be able to find general solutions or solutions to an IVP for the types of equations listed here. You should also have the abstract outline in mind that guides these methods: linear equation: I find an integrating factor, and to do this I solve a separable equation Bernoulli: the substitution (which?) converts it into (which simpler type?) homogeneous: -"- v=ax+by : -"- When you compare with the book, remember that you are NOT responsible for exact ODEs, for sec. 2.5, and for the method on p 77-78. The review problems of Ch.2 help with this discernment as well as training. The fact that some of the review problems fall in the categories you cannot do is an advantage in my opinion. You should be able to pinpoint them and discard them (and to assign the appropriate method to each of the others). I also remind you that the method for linear equations, as studied in Chapter 4 (superposition principle, undetermined coefficients, variation of parameters), do apply for first order as well, and they can be compared with how we did 1st order linear ODEs in chapter 2. This comparison is made in my lecture notes at the beginning of the notes for chapter 4. (The book omits such comparison). If you reread these notes from the distance in time, it will help you assemble some material more coherently, with *indirect* benefit on the exam, because what you learn will look less like a bag of separate tricks. From the modelling chapter, notice the mixing problems, Newton's mechanics, and the population models. (No, I won't ask you to memorize the names `Malthusian' and logistic'). But do remember the trick when we found hit-bottom times and ended up with equations like t + exp(-t)=10, which shouldn't baffle you. Don't worry about the dazzling distinctions about different heat sources they make in the book, but do remember simply Newton's law of heating and cooling. The ODE you get from this law is of course a very easy one. *If* I choose a mechanics problem, I vow NOT to go slugs and pounds (because in the supermarket, pounds refer to mass and slugs to a pest, whereas in the lab, pounds refer to a force and slugs to a mass, and it is a pest that the book never really sorts out this confusion). Remember that we skipped 3.5-3.7. The core material of Ch. 4 can be found in 4.2-4.6. But make sure that you also refer to the notes, because the most difficult part is to keep the overview, and the notes focus on how it all fits together, whereas the book sections focus on how each individual task is accomplished. A typical problem may be an IVP for a lin. inhom. ODE and you need all pieces together: - the exp(rt) trick to get the hom. solution (or I provide needed information in case it is not constant coefficients) - either undetermined coeffs or variation of parameters to get a particular solution - superposition principle to put them together and then (not sooner) use the IV's to find the constants of integration. Some of these problems can also be done by the LT method. If I do not require a particular method, you have the choice. (And I prefer not to prescribe methods, b/c all you learn would be no good if you needed someone to tell you the method. I *may* prescribe a method if otherwise I couldn't enforce covering this method in the exam) Make sure in which cases you need extra powers of t in an undetermined coefficients of homogeneous ODE example. If you can recall the idea how I explained the occurrence of the extra factor t as a limit where two roots are `almost' equal, that would be just great. But you won't be asked to redo the precise calculation. We skipped 4.7 at the time, however the principle explained from p.200 `The energy integral lemma' through example 2 is exactly what I did in the course notes a bit earlier, namely on page 8 ``an important integrating factor in mechanics''. It's irrelevant there that the rhs contains sin x; any other fct of x does the same trick. I do want you to know about this special `integrating factor' method: It's more important in `real life' than some of the substitution methods from Ch. 2. The basic complex arithmetic with Euler's formula is part of the required material. See the notes and some posted hwk solutions, which are more elaborate on this point than the book. You must be able to convert between trigs and complex exponentials. (Euler's formula). Remember that this skill includes knowledge that comes in handy when discerning which roots to look for (and which not), before you determine whether a form for undetermined coeff's requires extra powers of t. Also remember that it was the Taylor series for sin, cos and exp that produced Euler's formula for us. (Yes, you should be aware of this *fact*, but no, I am not asking you to redo the proof, even though, honestly, it's not that tough. Try it, then compare with book/notes, and then you won't have trouble remembering the fact as such.) For the resonance (damped forced oscillator), rely on the notes. I continue to require your understanding of the resonance phenomenon, including the phase shift, how it all depends on the forcing frequency and the damping, and to know a resonance when you see one. When is resonance destructive (how does it depend on frequency, damping)? For the discussion on resonance I skipped all the many calculational examples on critically damped and overdamped case, and the exam will abide by the same philosophy. If you want to get a better understanding upon review, take the resonance calculation from the lecture (it's in the notes as well), choose some numbers for m, k, b, (but leave gamma variable) copy it on one sheet of paper, then parallel it with your own calculation (on another sheet; it could be a hwk on pg 186) using sin and cos instead of the complex trigs I used, and compare. It will be less mystifying at second look than it was at first. Remember that in principle, all methods from Ch. 4 work for linear ODEs of higher order as well. We had a few examples, and they are fair game, provided the calculational complexcity is manageable. (Linear independence of solutions, and the Wronskian take a bit more discussion for higher order, but these won't be on the exam. You are however advised to review them if/when you are going on to M251, because that way you harvest synergy effects.) The book covers higher order material in 6.2 Exercises 1-14, 19-21, and 6.3, Exercises 1-10, 31-33, and 6.4, Exercises 1-5, but you should refer to my notes or the text from Chapter 4, NOT to the text from Chapter 6 (because the book introduces language there which we didn't cover). The examples from the notes are enough to describe the extent to which higher order might possibly be covered in the exam. (Remember that the focus is to teach you that these methods in principle work for *any* order, not to introduce more specifics for higher order.) Finally, you need to know the Laplace transform method and do direct and inverse Laplace transform, with all the partial fraction decomposition tricks and shortcuts I taught you. Have another look at the formulas that create new Laplace transforms from old (remember Pblm 3 on Exam 3). You need to know when to shift which variable in which direction. L{e^(at) f(t)}(s) = L{f(t)}(s-a) from pg 360 L{u(t-a) f(t-a)}(s) = e^(-as) L{f(t)}(s) from pg 387 L{u(t-a) g(t)}(s) = e^(-as) L{g(t+a)}(s) (and the variants for the inverse LT) Likewise you need to know the role of derivatives L{f'(t)}(s) = s L{f(t)}(s) - f(0) (and higher derivs) from pg 361-362 but also L{t f(t)}(s) = -d/ds L{f(t)}(s) (and higher powers) If you find this awful to memorize, you are right. Here are some guidelines to help. For each formula, have another look at the proof in the book. Think of taking a highlight marker and trace through these calculations just the parts of the formulas which eventually conspire to give the shift In particular those who haven't absorbed these formulas yet should think of the interplay power <-> derivative and exponential <-> shift in these formulas. But watch out that it is not quite symmetric; chase with the highlighter through the proofs (only a few characters per line get highlighted, namely those that trace features that show up in the final formula): if you can do this, you'll understand and memorize these formulas much better. I will again provide Table 1 for you. If you are wise, it will guide you in the memorization of the mentioned formulas, which I do NOT provide. Try this: ``I have forgotten the formula or am unsure about it, the book is closed, but I do have Table 1. Can I use it to reconstruct the forgotten formula?'' To make you think through this question is one of the rationales for again offering you Table 1. Table 1 is `half of a cribsheet' for the formulas, but only if you acquire the wisdom to read it as such. And I care more for this wisdom than for your memorization, so take advantage! Since the primary benefit of LT is when you have piecewise defined functions, including periodic, make sure you can handle them well, unit step function translation and all, direct and inverse LT. For periodic functions, I'll ask only direct LT; but be reminded that you *can* do ILT by returning to the geometric series. The geometric series and how it wraps up infinitely many shifted pieces into the (1 - exponential) in the denominator is required fact knowledge. Think of it as easily accessible partial info: ``If I see a piecewise defined function with certain `switch times', I still find these switch times in the LT of that function. Where? If it is a piecewise defined periodic function, this feature still shows up in a discernible place in its LT. Where? '' I want you to know these things because they serve as landmarks if you have a complex calculation. The exam calculations are not meant to be messy, but real life calculations are. This is why I force you to learn the landmarks for the exam. Look into the Q&A part of the notes on LT: Is every function somebody's LT? Can two functions have the same LT? What properties must *every* LT have? This is also expected fact knowledge. ******************************************************* I'll strive to have conceptual questions at 10-15% of the total, the rest calculational. Conceptual questions may be fill in the blank, or a brief free-response explanation, or the kind you have seen on in-class exams. They will cover clearly and repeatedly stated facts and should be answerable by reference to such. Go through the conceptual knowledge I listed here and try to come up with your own exam question. (E.g., read one snippet at a time, and try to concoct a question about it while sitting in the bathtub:-) In an earlier 231 class I had about 10 questions on the final. Actual numbers may vary slightly, depending on the questions. The idea is that the final, compared to in-class exams, should be *less* than twice as much work for a tad *more* than twice as much time. Of course you want to redo some hwk or exam problems, as needed. This is the obvious part, so I didn't bother dwelling on it.