Ma 341 section 2 Homepage

Welcome, please note that the schedule and syllabus are linked just below.

• Course Syllabus for Ma 341-002 ( grading scheme office hours etc...)
• Course Schedule ( suggested problems assigned and test dates )
• Errata ( errors found so far )
• Differential Equations covered in calculus II A synopsis of things you should have seen before. We will rehash much of the subject matter from Ma 241 as we cover the more general case in this course (n=2 or 1 in Ma 241, we cover n=1,2,3,4,...).
• Table of Laplace Transforms given for test III and final:

• Test Solutions from this Semester:

• Test I Solution: May 28, 2008.
• Test II Solution: June 11, 2008.

• Test Reviews and Practice Tests:

Keep in mind if I said something that differs from this review in class then what I said in class wins. If in doubt ask.
• Test I Review: overview of what's on the test.
• Practice Test I: problems similar to those on test.
• Practice Test I solution:
• Test II Review: overview of what's on the test.
• Practice Test II: problems similar to those on test.
• Practice Test II solution:
• Test III Review: overview of what's on the test.
• Practice Test III: problems similar to those on test.
• Practice Test III solution:

• Solutions to Tests from Fall 2007:

I intend to focus a little more on applications this time so these tests are only a partial representation of what you need to know. The same goes for the practice tests. I will likely add a few more applied problems throughout the course as I drop a few of the more advanced theory items.
• Test I solution:
• Test II solution:
• Take Home Test solution:
• Test III solution:

• Homework Problems and Solutions:

You should notice that only 10 points are required in the homework. This means that there are 5 points extra mixed in here. That means that in principle you can earn 105pts in the course overall. The reason for this is that certain homework problems are really really hard. I'll let you know which ones if you ask.

• Homework One (upto 5pts of total grade) DUE MAY 23
• Homework One Solution

• Homework Two (upto 5pts of total grade) DUE JUNE 6
• Homework Two Solution

• Homework Three (upto 5pts of total grade) DUE JUNE 18
• Homework Three Solution (will appear)

• Homework on n-th order ODEs and the basics. (like Test I problems)
• Homework on systems of differential equations. (like Test II problems)
• Homework on Laplace Transforms. (like Test III problems)
• Homework on Project Related Problems.

• These solutions contain a few errors, if you think you found one send me an email with a clear exposition of what the error is and how to fix it and you'll earn a bonus point on some test, this also goes for the notes. Of course once I notify the class of the error you may no longer ask for that point. To start with check our page of known errors (errata page).

Course Notes:

Lecture will often closely follow these notes. Your text covers most of what is in these notes, I add some material on the matrix exponential, the imaginary exponential and finally I add some comments to the energy analysis. Mostly the text has much more to say, Newton's Law of Cooling and the analysis of springs are much more verbose and detailed in your text and certainly worth reading if you're interested. I try to just stick to the math.

• Differential Equations in Physics:(no numbering) This is not required material, I mention just a few of my favorite equations in the hope that I can convince you that differential equations are essential parts of modern physics.
• Complex variables:(i-8i) While this course is mostly about real variables we will at times find ourselves using complex variables to solve a problem. I give a brief overview of the rules for complex variable manipulation, nothing surprising really, except perhaps the rule exp(ix)=cos(x)+isin(x). This fascinating relation allows us to derive any trig. identity we desire. My real intent with these notes is that you become a little more comfortable with the concept of a complex variable and also you learn how to use them to derive any trig. identity you might need later in this course.
• Introduction:(1-7) We define the basic vocabulary and discuss direction fields, Euler's Method and separation of variables. A table of contents links these notes to the corresponding sections in your text.
• First Order ODEs:(8-12) We introduce the integrating factor method to solve tricky 1st order ODEs that separation of variables cannot handle then define exact equations and how to solve those.
• Applications:(13-16) Mixing tank problems, Newton's Law of Cooling, and Newtonian Mechanics. All phrase problem as a differential equation.
• n-th order homogeneous ODE:(29a-29e) These are a shorter version of the longer 14-30 notes, here I focus on the sole goal of showing why the solutions given on page 30 are indeed solutions.
• n-th order homogeneous ODE:(14-30) Theory of linear ODEs, linear independence, the Wronskian, differential and linear operators all used to motivate the solution to the general n-th order linear ODE with constant coefficients.
• n-th order nonhomogenous ODE:(31-39) Extending our knowledge from the homogeneous case we learn two main methods to find particular solutions. The metod of undetermined coefficients asks us to "guess" a particular form for the particular solution which we then must determine the coefficients A,B,C,... by doing some algebra. The annihilator method explains those "guesses". Variation of parameters requires integration in contrast to undetermined coefficients and as such it is a weapon of last resort, it can of course treat more difficult examples at the cost of some hard integrations.

• ( Test I covers pages 1-39 and pages i-8i in my notes )

• Phase Plane:(pp1-pp18) We examine systems of two ODEs from a qualitative point of view. The Phase Plane is a tool which allows is to graphically tackle the form of solutions for such a system. Various types of asymptotic behaviour are discussed, we describe the possible motions for a stable or unstable equilbrium point. (this material is taken partially from chapter 5 and also the beginning of chapter 12. You will not be tested on this material in Tests I,II, or III, however you will be tested on it in the Project and also in the Final Exam) ,
• Two Equations, Two Unknowns:(72-74) Systems of equations can be surprising at times, but it all comes back to the cases that arise in this simple case. The methods presented in this section are useful for many problems in this course and elsewhere.
• what is the matrix ?:(75-82) Far from being a surreal expidition into philosophy-laden science fiction, the matrix is a notational convenience which allows us to solve many equations as if they were one single equation. The methods presented here are useful in many many many places.
• Theory for systems:(83-88) We see how systems of 1st order linear ODEs have a special position in the big-picture of things, any linear n-th order linear ODE can be viewed as a system of 1st order ODEs. On the flipside we can expect certain features from our experience with the n-th order case, indeed much of the theory is analogus.
• Simple Homogeneous Linear systems:(89-99) I say "Simple" because we examine only the cases where there are enough eigenvectors to comprise the solution. Interestingly, we can have a double root and still no factor of "t" appear since the presence of vectors in the solution introduces a new facet to the linear independence of our solutions.
• Nonhomogeneous Linear systems:(100-103) Again we discuss two methods to extract the particular solution for a nonhomogeneous linear system of ODEs. Undetermined coefficients is considerably more difficult in this context, however variation of parameters inherits an elegance from the matrix formalism which makes it more tractable in my limited experience.
• Matrix Exponential:(104-123) Here we learn how to solve the less than "Simple" cases. It is the case that exp(tA) is a fundamental matrix for the system of ODEs x'=Ax, but what does that mean? And how do we even calculate such a thing? We introduce the "generalized eigenvector" to aid us in this task, in the process we gain greater insight into why the eigenvector methods worked before. Eigenvectors are also useful in many applications besides the one under consideration in these notes, so experience in calculating eigenvectors is not a bad thing.
• Energy Analysis:(pp19-pp27) As we know any second order ODE is equivalent to a system of two first order ODEs in normal form. If we choose one of the variables in our system to be velocity then we find a system of ODEs in x and v=x'. In classical mechanics one calls the x,v plane the phase plane. We analyze how energy is a constant of the motion for conservative systems. The potential function G gives us a neat tool to construct the phase plane plot, essentially we can use calculus one min/max theory to find the critical points. After mastering the conservative case we move on to the more subtle nonconservative case (think adding friction to the mix). It turns out we can use the conservative case for a system to construct the solutions in the nonconservative case. (this material is taken partially from chapter 5 but mostly from chapter 12 you will not be tested on this material in Tests I,II, or III, however you will be tested on it in the Project and also in the Final Exam)

• ( Test II covers pages 72-123 in my notes )

• Laplace Transforms:(40-48) Basic definitions of the Laplace transform given. Then various theorems discussed that allow us to transform a differential equation in "t" to a corresponding algebraic equation in "s".
• Inverse Laplace Transforms:(49-54) We learn how to reverse directions. Given an algebraic expression in "s" we find a corresponding expression in "t" ( not usually differential since we solved the algebra )
• Discontinous functions and Laplace Transformations:(55-61) Here lies the real reason to discuss Laplace transformations. We can deal with discontinuous forcing functions in an easy manner that doesn't require us to break up into cases (like we would have to otherwise )
• Convolution:(62-65) An interesting concept which gives us another tool for performing inverse Laplace transforms.
• Dirac Delta Function:(66-69) Probably the weirdest thing in the whole course, you'll see.
• Laplace Transformations and Systems of ODEs:(70-71) We follow same idea as in the case of just one equation, difference is that instead of getting a single algebraic equation to solve whereas for a system of differential equations we get a system of algebraic equations to solve. Then of course we have to take the inverse transform to finish the job.

• ( Test III covers pages 40-71 in my notes )

Beyond the notes: using software to explore DEqn's

You have a 5 point project that requires you to use Matlab to analyze certain ODEs. The "pplane" package is capable of plotting excellent phase plane plots. We will use it to supplement our understanding of Chapter 12 and 5 material. See the course schedule for more details.

Tutorials from Fall 2007 students Your project is different, but the advice on "pplane" is still relevant.
• Windows based tutorial ( need Java, may not work on school computers ):
• Linux based tutorial ( should work on school computers ): 