Here are solutions for the practice problems. A few of the problems were not adequately checked by me, especially 11c, 12, and 13. For 12 and 13, you might practice setting up the differential equation, but do not worry about solving them.
I've put together some review problems for your final. The actual final is Wednesday, March 14, 2012, at 8:30 in our room. The review word problems have discontinuous and Dirac forcing terms, something we will spend the rest of this week learning about.
This is your final homework assignment, due Friday. We'll finish covering impulse functions on Monday, and then we'll spend the rest of the week reviewing and doing applications of the new Laplace transform techniques.
| Section | Assigned Problems | Suggested Problems |
|---|---|---|
| 6.4 | 9, 10 | |
| 6.5 | 4,5 |
Our penultimate homework assignment will cover how to use Laplace transforms to solve differential equations and a little theory. The problems from section 6.1 ask you to determine the convergence or divergence of a few improper integrals. We'll cover some more of this in class on Monday, as I backtrack to cover some of 6.1.
I'd like you to be aware that the definition of piecewise continuous I provided you in class does not exactly match the definition in the book. The book requires an extra condition, which for us boils down to a piecewise continuous function can have no asymptotes. I will continue using the simpler definition I gave you in class, and I'll give some theorems that are slightly more powerful than the one Boyce and DiPrima give.
The homework is due next Friday, March 2, 2012.
| Section | Assigned Problems | Suggested Problems |
|---|---|---|
| 6.1 | 21,23 | |
| 6.2 | 4,7,14 |
I've posted Quiz 4 solutions for those who would like to check their work in preparation for the upcoming midterm. Understanding how to get the correct differential equation from the text is an important part of the problem!
Shuwen Lou, another grad student here, has kindly made her Midterm practice exam available. It provides a good idea for the type of exam I'm planning to give next Wednesday. Here is her test, and here are the solutions.
Undetermined coefficients will be a large part of the upcoming midterm (February 22). To help you prepare, I've put together some practice problems together with a short synopsis on the technique. The solutions are available to see if you're on the right track.
Homework 5, covering material from Section 3.7, is up. Looking ahead to next week, we will be having a quiz on the Friday a week from tomorrow. Because the week coming after that (two weeks' time) has a midterm and a holiday, this homework will be due the following Friday.
| Section | Assigned Problems | Suggested Problems |
|---|---|---|
| 3.7 | 7,11,17,21 |
Homework and test make-up points are both due next Wednesday, February 8, 2012. Remember that the second bonus homework is also due then. I changed my mind, and I have decided to make it possible to gain points back for the matching problem, but this will be done differently than the other problems (details are provided further down). You may choose to re-attempt two problems total.
Here are the guidelines for the written make-up points (Problems 1,2,3, and 5). Write out the solution to the test problem completely on a different sheet of paper. If it is a multipart problem, clearly label which parts of the problem are which. Turn in the problems together with your test on Wednesday. I will only award make up credit for completely correct second attempts, and if I award credit, then I will award half of the points missed on the problem (including the fraction).
To get make-up credit on Problem 4, I will offer a make-up matching problem on Monday or Tuesday. You will need to take it in my office, and you will have 10 minutes to do it. Please arrange a time with me ASAP to do this. As with the other test problems, I will only offer make-up credit for a completely correct answer.
Office hours on Thursday will again be moved to Tuesday 4:30 for next week. Please make an appointment with me if you are unable to attend normal office hours.
With regards to the homework, I will make a few comments about second order differential equations without constant coefficients on Monday (not too much can be said), and this will help with the problem from 3.3. Using the suggested Problem 34 that precedes it, however, you will be equipped to solve it.
| Section | Assigned Problems | Suggested Problems |
|---|---|---|
| 3.3 | 35 | 34,36 |
| 3.4 | 18 | 24,27,28 |
| 3.5 | 13,15 | 5,7,8 |
I want to give a few extra comments on Problem 27(a) and (b), to help everyone get started. We want to set up a differential equation that regulates the velocity of the falling ball in a fluid. To do this, we will (1) find all the forces acting on the ball, we will (2) sum the forces on the ball, and we will (3) apply Newton's law of motion (\(F = m\cdot a = m \cdot v'\), with \(m\) the mass of the ball) to find the differential equation. In this problem, I'll take the convention that \(v(t)\) is positive when the ball is moving down.
One of the forces, \(R\), the frictional force due to viscosity, is given to us in the problem. The problem tells us that \(R = 6 \pi \mu a |v|\), but this is really a typo. It should read \(R = -6 \pi \mu a v,\) corresponding to a force that always pushes opposite the direction of motion.
The weight of the ball is the force due to gravity, and so this force is \(w = m g,\) where \(g \approx 9.8 m/s^2\) is the acceleration due to gravity. The last force is the Buoyant force, which the problem tells us is 'equal to the weight of the fluid displaced by the object.' This weight is given by \( m' \cdot g,\) where \(m'\) is the mass of the displaced water. The buoyant force pushes against the weight of the ball, and so \(B = - m' g.\) By adding the forces, we get that the total force on the ball is \[ m \cdot v' = F = R + B + w = -6 \pi \mu a \cdot v - m' g + m g. \]
Instead of telling us the masses of the objects, the problem tells us the densities. Therefore, we'll write the masses in terms of density and volume. The volume of the sphere can be computed in terms of its radius \(a,\) so \(m = \rho \tfrac{4}{3} \pi a^3\) and \(m' = \rho' \tfrac{4}{3} \pi a^3,\) where \(\rho\) and \(\rho'\) are the densities of the ball and medium respectively. Substituting these in for \(m\) and \(m'\) will allow you to compare with the back of the book.
Part (b) of the problem asks you to introduce a new force \( E\cdot e\) where \(E\) is the magnitude of the electric field and \(e\) is the charge of the particle. It tells us that \(E\) has been chosen so that droplet is at rest, i.e. \(v = 0\) and also \(F = 0\). Adding the \(E \cdot e\) term to the sum of forces above, and making these substitions, we can solve for \(e\) in terms of \(E\).
Woohoo! Here's the new bonus problem, which attempts to explore a more complicated population model. This should be lots of fun! There are actually three bonus problems in this document, but two of them are only worth half a bonus point (1% grade boost per bonus point). Since they're all connected, I'll have them due all at the same time: Wednesday, February 8.
Because of the snow days, we're going to have to accelerate our coverage of the some of the material. At a glance, here are the key changes to what we'll be doing for the next week.
Your next homework assignment is posted below. We ought to have everything you need to complete the homework by the end of this week. It looks a little longer than most, but you have a little longer to do it, and you'll find that the problems from section 3.1/3.3 are relatively quick.
There will also be a bonus problem, which will expand upon section 2.5. This will be due in two weeks, and I will post it after Monday's class.
| Section | Assigned Problems | Suggested Problems |
|---|---|---|
| 2.3 | 27 | |
| 2.4 | 22 | 7,8 |
| 2.5 | 18,20 | 3,4,7,16 |
| 3.1/3.3 |
Solve the following inital value problems:
|
3.1:9,10 and 3.3:7,17 |
This is your second homework assignment, due Wednesday, January 18, 2012. There will be a bonus problem as well, but I haven't had a chance to say anything about the first bonus. Because of this, the bonus problem from last week will be due next Wednesday.
| Section | Assigned Problems | Suggested Problems |
|---|---|---|
| 2.1 | 13,16,20 | 1,2,3,18,19,28 |
| 2.3 | 12,18ac | 2,6,8,32 |
Some of you have asked me if it's possible to use an old version of the book. I've looked into it, and I discovered two interesting things. First, the syllabus says we are using the 10th edition. That's strange, because the 10th edition doesn't exist yet. In fact, we are using the 9th edition, and this is the second class I've taught where I was using space-and-time defying 10th edition. Second, it is possible to use the 8th edition of the book. Hart Smith has posted a list of differences here.
On the subject of defying space-and-time, here are the solutions to the review integrals. I apologize for not getting them to you sooner. I'm also posting the quiz solutions. These will be returned on Monday.
The first homework assignment is due next Wednesday in class, and the Bonus Problem is due at the same time. Please remember to staple your homework.
Homework is essential to learning mathematics; arguably, the only way to learn math is to do it. That said, I've made the homework assignments relatively short, to allow you to spend time preparing for the class as you see fit. For this reason, I've supplied additional problems that provide you with extra practice if you desire to do them. These will not be graded, but I encourage you to discuss them with me in office hours.
| Section | Assigned Problems | Suggested Problems |
|---|---|---|
| 1.3 | 17 | 14,19 |
| 2.2 | 13ac, 14ac, 22, 23 | 2,3,6,8 |
The first bonus only has a simple computer part, but this will give you a chance to familiarize yourself with a computer system. You'll want to refer to problem 2.2.30 and the blurb above problem 30 for clues and definitions. Bonus problems are due the same day as regular homework.
Hello and Welcome to the class webpage for Math 307 I. Our class resources, including homework assignments, will be posted on this page as we progress through the class. While you're here, be sure to glance at the syllabus, the preliminary calendar, and, if you're interested, the short Maple introduction.
| Office | Padelford C-8K |
|---|---|
| paquette//math.washington.edu | |
| Office Hours | Thu 4:30-5:30, and Mon 10:30-11:30 |
| Lecture | MWF 9:30 - 10:20 in LOW 205 |
|---|---|
| Midterms | Wed Feb. 1st and Wed Feb. 22nd |
| Final | Wed Mar. 14 in LOW 205 |