Math 2410Q-005, Fall 2012
If only one set of section numbers is given, it applies to both the third and the fourth editions. If the section numbers differ, the section numbers for the third edition are the ones in red, and the section numbers for the fourth edition are the ones in blue.
This class gave us some basic practice in setting up the kind of problem we'll keep seeing in this course. One thing to keep in mind is the difference between dy/dt and y(t): If a problem asks you about whether a quantity is increasing or decreasing, you need to know if dy/dt is positive or negative. If they ask you whether the quantity itself is positive or negative, you need to know about y(t).
|Week ||Sections in text (estimated) ||Quiz/Test ||Administrivia|
|Week 1: Aug. 27-31 ||1.1, 1.2 || |
|Week 2: Sept. 3-7 ||1.3-1.5 ||Quiz #1 ||No class on Monday (Labor Day)|
|Week 3: Sept. 10-14 ||1.5-1.7 ||Quiz #2 ||Monday: last day to drop without a W |
or change grading option to pass/fail
|Week 4: Sept. 17-21 ||1.7-1.9 ||Quiz #3 |
|Week 5: Sept. 24-28 ||1.9 ||Midterm #1 |
|Week 6: Oct. 1-5 ||2.1, 2.2 ||Quiz #4 |
|Week 7: Oct. 8-12 ||2.3, 2.4, 3.1 |
|Quiz #5 |
|Week 8: Oct. 15-19 ||3.1, 3.2 ||Quiz #6 |
|Week 9: Oct. 22-26 ||3.3 ||Midterm #2 |
|Week 10: Oct. 29 - Nov. 2 ||3.4, 3.5 ||Quiz #7 ||Monday: last day to drop |
or change a pass/fail option to letter grade
|Week 11: Nov. 5-9 ||3.6, 4.1, 4.2 ||Quiz #8 |
|Week 12: Nov. 12-16 ||4.2, 4.3 ||Quiz #9 |
|Week 13: Nov. 19-23 || || ||Thanksgiving break!|
|Week 14: Nov. 26-30 ||6.1-6.3 ||Quiz #10 |
|Week 15: Dec. 3-7 ||5.1 and review ||Quiz #11 |
The problem I gave you at the end of class was this: you have a 1000 gallon tank that catches runoff from manufacturing. At time t=0, it has 800 gallons of water with 2 oz. of pollution in it. Polluted water (containing 5 oz. of pollution per gallon) enters the tank at a rate of 3 gallons per hour. The contents of the tank are well mixed and exit the tank at a rate of 3 gallons per hour. We set up a differential equation based on these facts, and I asked you to solve it at home.
Here's an additional complication: now suppose that when the amount of pollution reaches 500 oz., the polluted water is cut off from entering the tank. At that point, fresh water starts entering the tank at a rate of 2 gallons per hour, and the well mixed contents of the tank now exit the tank at a rate of 4 gallons per hour. Can you calculate (1) when the amount of pollution will reach 500 oz. and (2) set up the differential equation showing what happens after that point?
There's a nice Java applet for Euler's method here. Try it out with the problem we did in class or the following initial-value problem: dy/dt=y+2yt, y(0)=1, 0<=t<=2, delta t = 0.5. This is a separable equation, so you can solve it yourself and compare the actual value to the approximation you get using Euler's method. Is it a better or worse approximation than the one we got in class? Why do you think that is?
You can find good resources for working with slope fields on this website. It has practice exercises, including one on matching slope fields with the corresponding differential equations, and an applet you can use to draw slope fields and add approximations using Euler's method. And now that you've had today's class, you know why the solutions you draw in the slope field shouldn't cross!
The trickiest thing about phase lines seems to be remembering what each function represents. We're given information about dy/dt (either a function or its graph), so the y-line we draw has information about whether the slope of the solution is positive or negative. Then, when we sketch the solutions, we're drawing possibilities for the solution y(t).
Today's note is just a quick terminology review: the phase line is the vertical y-line with the arrows on it indicating whether dy/dt is positive or negative. This is not the same as the sketch of all possible solutions you can create from the phase line. (For example, in the third edition, Figure 1.62 shows a phase line and Figure 1.62 shows the corresponding sketch of solutions. In the fourth edition, Figure 1.64 shows the phase line and Figure 1.65 shows the corresponding sketch of solutions.)
I strongly suggest that you take a look at the book's treatment of the example I finished class with. Bifurcation points are not the simplest thing we've dealt with so far!
We spent the day talking about the "lucky guess" method in Section 1.8. Most other books will call it the "method of undetermined coefficients." The general idea is that you can identify the form of a solution to the nonhomogeneous equation based on the form of b(t). The only thing you won't know are the constants involved. All you have to do is plug that form into the differential equation with arbitrary constants and then solve for the constants!
You should also check that you need to multiply the standard form by t for the last example we did in class. Try using the standard form and see what happens when you plug it into the differential equation!
Here's the extra example I promised you: Solve the initial-value problem dy/dt = (3t^2)y + 18e^(t^3)), y(0)=2.
As we move into systems of differential equations, more of the material you learned in calculus class about parametrizing curves and vector functions will be useful. If you haven't seen it in a while, you might want to review it over the next few days!
If you'd like to try drawing direction fields of some systems, try using the applet at this site. It's got some preloaded examples, and you can enter your own. Try some of the examples from the textbook or the examples we talked about in class. What happens if you change the value of k/m in the damped harmonic oscillator example?
Today we're continuing with our theme of doing for systems of first-order equations what we've done for single first-order equations. Try out Euler's method on the system for the damped harmonic oscillator! If you'd like to try it with a Java applet to check your work, you can use this one. You should also notice that we don't need to know the value of t that gives us the point (x_0,y_0). Why do you think that is?
You can study matrix multiplication here or with this Khan Academy video. You'll have to be able to understand this for this course, so please practice this if you think you need it!
Think a little bit about how linear independence relates to whether you can solve an initial value problem in a 2D linear system! What conditions do we need on the solutions we're given?
We spent quite a lot of the class talking about straight-line solutions. Given that we're working with autonomous systems, what do these solutions tell us about the others geometrically?
We've talked about solving linear systems using eigenvalues and eigenvectors today. Use these techniques to solve the partially decoupled systems we've solved using other methods and check your work!
Note that when we solved the system for which we got complex eigenvalues, we only needed to look at one of the eigenvalues and its associated eigenvector to find our general solution. You should go back and find the general solution using the other eigenvalue and convince yourself that you get the same thing.
If you're not confident in your ability to add and multiply complex numbers, this Khan Academy video might help.
This would be a good time to start organizing your knowledge of the different kinds of solutions of linear systems. You could work out a chart where each row corresponded to one type of pair of eigenvalues (distinct and both negative, distinct and one positive and one negative, repeated real eigenvalues, etc.) and contained a sample system with that type of eigenvalues, a possible sketch of a solution to such a system, and the general form of the solution.
Think about how the direct method of solving constant-coefficient homogeneous second-order linear equations in section 3.6 is related to our method for solving them by converting them to a system of first-order linear equations. How is the polynomial we get using our new method related to the polynomial we get when we convert to a system and find the eigenvalues, and why does that relationship hold?
Here's an applet to play around with. You can input values for k, m, and b and watch what happens with a spring for those values. And as the spring moves, a graph of the solution will be generated! Go have fun with it!
I recommend trying some exercises from section 4.2 using both the method of undetermined coefficients and the method we used in class today. You can check your answers that way, but remember that you can also always check your answer by plugging your solution into the original equation and seeing if it works!
Here's the promised footage of the Millennium Bridge in London. The explanation of resonance is quite good.
We'll start talking about Laplace transforms on Thursday. It's a very cool computational technique! This is what we'll need partial fractions for, so you might want to review those in advance.
If you've got the fourth edition, the Laplace transform tables are on p. 626. If you have the third edition, they're on p. 620. I recommend taking initial-value problems from sections 1.8 and 1.9 and solving them using Laplace transforms to get a bit more practice.
Here's a link to the newsreel footage of the Tacoma Narrows Bridge collapsing. If you want to read more about it, Section 4.5 of your textbook (both editions) has a very nice writeup.
You should have noticed by now that if you have a table of Laplace transforms, no calculus is necessary to solve differential equations. The main tricks you need to know are (1) how to do partial fractions and (2) how to use that table. Please practice both!
We got through the sections in 6.3 called "Laplace Transform of Sine and Cosine" and "Shifting the Origin on the s-Axis." The rest of the section is full of examples of Laplace transform problems and might be useful to you.
I'm sorry I didn't get to the examples of setting up differential equations to model a situation. If you want to practice that, I recommend #17 in section 1.1 and ##9-14 in section 2.1.
Good luck as you all study for your finals! It's been fun teaching you.