Math 303, Fall 2018

Final exam solutions

Here they are.

Midterm 1 review materials

The practice midterm. Not proofread -- please let me know if you find any mistakes!
My review sheet summarizing what we've covered so far.
Paul's Online Math Notes (no relation) about systems of differential equations (doesn't cover nonlinear systems).
The first section of this has some great and original examples of modelling phenomena with ODEs.
Boyce and DiPrima's Elementary Differential Equations is another textbook covering the same material that you might find easier to read. You can easily find a free PDF of this.

Midterm 2 review materials

Here is a practice midterm, which we will go over in class on Tuesday.
I didn't make a review sheet, but the topics covered are nonlinear mechanical systems (section 6.4), finding Fourier series (9.1), the convergence, differentiation, and integration theorems (9.2), sine and cosine series (9.3), solving nonhomogeneous ODEs with Fourier series (9.4), and solving the heat equation (9.5).
Boyce and DiPrima is still a good substitute for our textbook. You need to find the version called Elementary Differential Equations and Boundary Value Problems.
Another nice book, which is freely available online, is Lebl's Notes on Diffy Qs.

Final review materials

Here's my review sheet, summarizing what we've covered in this class.
I think that the best way to prepare for this exam is to do problems from the book, by hand, focusing on areas you're less comfortable on.
Here's the practice final!
Solutions to the practice final.

Schedule

(This is all subject to change.)

Date	Sections covered	Homework due	Notes
8/20	5.1-5.2		Course policies, review of linear algebra, solving a linear system of 2 ODE's.
8/22	5.1-5.2		More linear systems, the eigenvalue method, complex eigenvalues.
8/27	5.2-5.5		Complex eigenvalues. Higher-order equations are first-order systems. Brief look at pplane.
8/29	5.5	Online 5.1 and 5.2; Homework 1	More complex eigenvalues, and repeated eigenvalues.
9/3			No class!
9/5	5.3	Online 5.5 and Homework 2	The relationship between eigenvalues and the behavior of solutions of linear systems; graphing with pplane.
9/10	6.1		Understanding eigenvalues in higher dimensions. Nonlinear systems: general properties, how to find critical points, and stability of critical points.
9/12	6.2	Online 5.3 and Homework 3	Linearizing nonlinear systems. The Jacobian.
9/17	6.3		Applications of nonlinear systems: predator-prey and competing species.
9/19	6.3	Online 6.1 and 6.2 (two parts), and Homework 4	More about predator-prey and competing species. Solving ODEs in MATLAB with ode45.
9/24	Midterm review!	Bring in a question or problem you want to talk about.	We'll go over the practice exam and discuss any other questions or problems people have.
9/25			Midterm 1 in ME 1061, 8-9 PM.
9/26	6.4		Nonlinear (soft and hard) springs, and the nonlinear pendulum. We introduced the useful technique of, given a system of diff eqs for x and y, solving for dy/dx and integrating to get y in terms of x.
10/1	9.1		Introduction to Fourier series, and how to compute them. Here's my graph of a square wave, and how a variable-frequency oscillator responds to a square-wave force.
10/3	9.2	Online homework 6.3 and 6.4, and Homework 5.	We went over the midterm. We used Fourier series to solve an ODE consisting of a harmonic oscillator forced by a square wave. Convergence of Fourier series for piecewise smooth functions. Fourier series for general periodic functions.
10/8			No class!
10/10	9.3	Online homework 9.1 and 9.2, and Homework 6.	More solving differential equations with Fourier series. Termwise differentiation and integration.
10/15	9.3		Even and odd functions, and Fourier sine and cosine series. Forced harmonic oscillators and resonance.
10/17	9.4	Online homework 9.3 (part 2), and Homework 7.	Endpoint value problems. Forced damped oscillators.
10/22	9.5		We began discussing the heat equation and separation of variables.
10/24	9.5	Online homework 9.3 (part 1) and 9.4 (two parts), and Homework 8.	More on the heat equation. We looked at two sorts of endpoint value problems: ends held at zero, and ends insulated (so u_x = 0).
10/29	Midterm review		We'll go over the practice midterm, and any other questions you have.
10/30			Midterm 2 in MATH 175, 8-9 PM.
10/31	9.6		Brief discussion of the heat equation when the ends are held at a fixed nonzero temperature. We introduced the wave equation, and solved it for a string with fixed ends with initial velocity zero.
11/5	9.6		Continued talking about the wave equation, including the case where there's a nonzero initial velocity; the d'Alembert solution; and applications to music.
11/7	9.7	Online homework 9.5 (two parts) and Homework 9.	Methods for solving PDEs: separation of variables, splitting a problem into simpler ones, finding particular solutions for nonhomogeneous boundary conditions. We started discussing Laplace's equation, aka the steady-state heat equation.
11/12	9.7		We went over midterm 2, and solved the steady-state heat equation on the rectangle.
11/14	9.7-10.1	Online homework 9.6 (two parts) and Homework 10.	We solved the steady-state heat equation on the disk, and started discussing Sturm-Liouville theory.
11/19	10.1
11/21	10.2	Online homework 9.7 (2 parts), and Homework 11.
11/26			No class!
11/28			No class!
12/3	10.3
12/5	exam review

Course Policies

In-class expectations: I expect you to come to class and to read the relevant sections in the book ahead of time. If you can't make it to class, you should try to get good notes from someone who was there. I might say something important in class that isn't in the book or on the website. I also expect you to pay attention. It's all right to use a computer to take notes and so on, but if you know it makes it harder for you to pay attention, please put it away!

Grade calculation: Grades are calculated out of 600 points, made up of 200 points for the final exam, 100 points for each of 2 midterms, 100 points for online homework, and 100 points for handwritten homework.

Homework: Handwritten homework is due in class at the beginning of class on Thursday. Online homework is due at 11:59 PM on Thursday. Late homework will not be accepted. However, I will drop your lowest two hand-graded homework scores, and your lowest three online homework scores.

See the Quick MyMathLab Student Guide for using the online homework system. Our department just started using this system and we're still figuring things out -- please come talk to me if you have any problems with it!

The handwritten homework will generally be more involved than the online homework. It may be additional problems from the textbook, problems I write myself, or a project involving computer analysis of some equation or problem. Please, staple your homework and write or type it neatly. If you're typing it, and especially if you're planning on going deeper into math, you might want to look into learning LaTeX, or use LyX.

A solution to a math problem -- if it's more than just a calculation -- is an argument. You're trying to convince someone (me, or the grader) that you understand what you're talking about, and that your answer is right, against all possible objections. Presentation matters for this! In particular:

Use complete sentences, and break your answer up into paragraphs if it is long or has multiple lines of thought.
If you introduce a new variable, say what it is. ("Use the substitution u = tan θ" or "Let Z be the population of zombies.")
Don't use arrows to go from step to step. An arrow can mean a lot of different things. Use linking words like "Therefore," or explain what you're doing, like "Now we substitute this in for y in the previous equation.
Refer to theorems that are in the textbook, and ones that we talked about in class. You're totally welcome to look at other sources, but if you use them for your homework, you should (a) cite the source, and (b) explain the method or argument, in your own words, if it's substantially different from one we covered in class. (Part of what you're trying to do is convince me you know the material!)
If your answer uses a graph or a table, make sure it's clearly labelled and legible. If you make it on a computer, you can type the problem and include the figure inside it, or print it out on a separate page, label it "Figure 1" or whatever, and refer to it in your handwritten work.
You're encouraged to work with other people. Math is more fun if you do it together. Please write up your work yourself, in your own words, and if you worked with other people, give them credit at the top of your assignment. ("Collaborators: David Hilbert, Emmy Noether" is a fine way to do this.)

Exams: There will be two midterms and a final. No calculators or notes will be allowed. There will be no make-up exams except in extremely unusual circumstances -- please talk to me as soon as possible if you think you might be in such circumstances.

Midterm 1: Wednesday, 9/25, 8-9 PM, ME 1061.
Midterm 2: Wednesday, 10/30, 8-9 PM, MATH 175.
Final: 2 hours, date and time TBD.

Calculators: None of the exam problems will require a calculator or a computer. Homework problems may require you to do something on a computer (e.g. make a graph or slope field, or use MATLAB to solve ODEs numerically.)

Accomodations for students with disabilities: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247. In this mathematics course accommodations are managed between the instructor, student and DRC Testing Center. If you have an accommodation memo from the DRC, come see me as soon as possible so we can discuss your accommodation.

Diversity and inclusion: Purdue's official nondiscrimination policy statement applies to this class. In the context of a math class, what this means is: racism, sexism, homophobia, etc. against your fellow students will not be tolerated; I hold myself to the same standard as your instructor, and in particular I will evaluate you in a nondiscriminatory way; and moreover, we're all struggling together through learning differential equations, which is a pretty incredibly difficult thing when you get down to it, and we should do our best to treat each other with kindness and tolerance during this struggle.

Links and Resources

I'm fond of the online graphing calculator Desmos.

Wolfram Alpha and Symbolab can handle many hard integrals and matrix computations. As a general rule, I expect you to be able to do all the integrals you learned in calculus, to be able to do 2x2 matrix computations by hand, and to at least know what's going on for harder matrix computations. On the homework, though, you are more than welcome to use computational tools like these to handle more tedious or intractable computations.

You can find pplane, a Java tool for studying systems of ODEs, at this page. It only works for systems of two equations, only those that are time-independent, and has the usual limitations of numerical ODE solvers, but it is pretty useful for visualization and qualitative analysis. It's pretty old but the Java version works fine. The MATLAB version, I think, has problems on newer versions of MATLAB.

There is a decent 3d grapher at GeoGebra.

I've put all the MATLAB code I've used so far in class at this page.

Here's the Desmos graph of a plucked string I talked about in the context of the wave equation.

Math 303: Differential Equations and Partial Differential Equations for Engineering and the Sciences

Course Information

Homework