References are to the text by Stephen W. Goode, except when otherwise specified.

You may move to a particular week by clicking on its number: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
 

Solutions to Exercises. When a solution or partial solution to an Exercise is posted, it will be at the same address as the Exercise itself.

Old tests. Here are the second and the fourth hour tests and the final exam that I gave when I taught Math 2090 in Fall 2000, in case you would like to see them. These old tests are not necessarily representative of the tests that I'll give this semester.
 

We will cover a good bit of Chapter 1, with special emphasis on slope fields (also called direction fields), separable equations, and linear equations. Read Sections 1.1 - 1.4, 1.6, and 1.7. On your own, do these problems:

• 1.1: 1, 7, 9, 11.
• 1.2: odds 1-19, 35, 37.
• 1.3: 1, 3, 15.
• 1.4: 1, 3, 4, 5, 17.
• 1.6: 1, 3, 19.
• 1.7: 1, 3.

You will see repeated mention of the exponential function and the differential equation y' = ky. You should review the first-year calculus material that concerns the exponential and its inverse the logarithm. The exponential function can be defined by its differential equation, and all its main properties derived from it; here is a one-page PDF file showing how that's done.

Lecture topics: Direction fields (also called slope fields); Newton's Law of Cooling problems; equations of logistic type, with an example from chemistry; the integrating-factor method for first-order linear equations; the separation-of-variables-method; the equations for mechanical (or electrical) vibrations. You will find notes on vibrations equations at http://www.math.lsu.edu/~mcgehee/2090/OneSpring.pdf and http://www.math.lsu.edu/~mcgehee/2090/TwoSprings.pdf

Problem Set 1, due Monday, August 30: The first problem set will consist of problems A-F at http://www.math.lsu.edu/~mcgehee/2090/Exe-L1.pdf. A reasonable minimum objective is to get a rating of 7 or better on three of the problems. If you attempt a problem and get less than 7 on it, it will not count. So the best thing to do is to attempt all 7. It's advisable to do the reading suggested above, very thoroughly, and to work out the "on your own" problems. You may need to learn or re-learn some material from your calculus text.
 

Over the next two weeks, read carefully Chapter 2 through 2.4, plus 2.6 and 2.7. The introduction on pages 106-107 is a good statement of what's going on in this Chapter, which deals with first-order linear DEs.

• In 2.1, understand the definitions and theorems, especially the ideas of dependence and independence, and the Wronskian.
• 2.2: Reduction of order.
• 2.3: Constant-coefficient equations. They apply to one-spring vibrating systems.
• 2.4: The method of undetermined coefficients, also called judicious guessing.
• 2.6: Mechanical vibrations.
• 2.7: RLC circuits.

• 2.1: 1, 3, 5, 7.
• 2.2: 1, 3, 5, 7.
• 2.3: 1, 3, 5, 9.
• 2.4: 1, 3, 5, 9, 21 (which is an application to RLC circuits).
• 2.6: 1, 3, 11, 13.

No class on September 6; it's the Labor Day holiday.

Problem Set 2, due Wednesday, September 8: Do no more than seven problems out of the ten listed as Exercises G through P at http://www.math.lsu.edu/~mcgehee/2090/Exe-L1.pdf. Your minimum objective should be to do three with a rating of 7 points or better. Remarks about some of the problems:

• J is one of three types of problems, discussed in Section 1.12 of the text, which are really first-order problems in disguise. You may wish to read that Section. Example 1.12.1, worked out in the text, is like J.
• K asks you just to set up a second-order linear homogeneous DE. The physical problem is just another undriven mechanical vibration. Is there any damping? A similar problem is one of the exercises in the text.
• L,M are best done after a careful reading of Section 2.6.
• N,O: Using the method of Section 2.3, we find that each of these problems leads to a polynomial of degree 4 in r, say, in which there is no r-cubed or r term. Thus the polynomial is a quadratic in r-squared which can be solved using the quadratic formula; and for each value of r-squared, there are then two possible values for r.

Re-read the Sections of Chapter 2 that were recommended above, plus 2.8 and 2.9. You may wish to review what's in your calculus book on the basics of complex numbers, which our text reviews briefly on pages 173 and 174. You will need to know Euler's Formula, and to be comfortable with the quadratic formula when it produces non-real roots of a quadratic.

We are continuing with Chapter 2 in Goode's book. A second-order constant-coefficient differential equation can be the mathematical model for a mechanical vibrating system involving (say) a mass hanging from a Hooke's Law spring, possibly with damping and/or a driving force. A pair of such equations can model a system with two masses and two springs. We are considering various methods for finding solutions to such DEs (whether or not they arise from vibrating systems), including the method of undetermined coefficients and the method of variation of parameters.

Those same mathematical models can describe a one- or two-loop electrical circuit involving inductance, resistance, capacitance, and/or an emf device. For now, we'll restrict attention to mechanical vibrations.

Even so, you will wish to look at the book's presentation of RLC circuits. Students who wish to polish their understanding of circuits might like to read parts of the extensive treatment in a physics text like Fundamentals of Physics, Extended, fifth edition, by Halliday, Resnick, and Walker, published by John Wiley & Sons, Inc. In Math 2090, our primary concern is to deal with the mathematical models, but all mathematics students should be aware of the applications, since they may sometimes provide helpful physical intuitions and visualizations.

On September 13, I discussed the method of variation of parameters and worked an example. On this method, read Section 2.8 and do problems 1, 3, 5 in that section on your own. The method is called for in the next problem set, Exercise W.

I also solved an undriven, undamped two-mass two-spring problem by reducing the two second-order DEs into one fourth-order DE - this might be called an elimination method. We saw that every solution to this problem is a combination of two simple motions, one in which the two masses always move in the same direction and one in which they always move in opposite directions.

LSU was closed on September 15 because of Hurricane Ivan.

This week I'll talk about linear algebra. Read and study Chapter 3 in detail. Some of the important topics are:

• Matrices, matrix notation, operations with matrices, interpretation of a matrix as a linear map from one vector space to another, the range and kernel of such a linear map.
• Systems of linear equations, ways of solving them, the matrix of coefficients, significance of its determinant (when the matrix is n-by-n), the augmented matrix.
• The inverse of a square matrix; when it exists.
• Elementary row operations: Each of them may be accomplished by multiplying the augmented matrix on the left with a certain nonsingular matrix.
• Gaussian elimination, which leads to the Row Echelon Form (REF).
• Gauss-Jordan elimination, which goes farther and leads to the Reduced Row Echelon Form (RREF). When this is done, you can simultaneously find the inverse of the matrix of coefficients if it exists.
• Finding the solution when there's one or more free variables.
• Eigenvalues and eigenvectors.

Readings and Exercises to do on your own:

• Read Section 2.8; do 1, 5, 7. Expect a problem using the method of variation of parameters to be on Test 1.
• Read Section 3.1; learn the terminology; do 3, 7, 11, 15-22.
• Read Section 3.2; do 1, 3, 5, 11, 13, 17, 19.
• Read Section 3.3.

Problem Set 3, due Wednesday, September 22: Do no more than seven problems out of the ten listed as Exercises Q through W at http://www.math.lsu.edu/~mcgehee/2090/Exe-L2.pdf. Your minimum objective should be to do three with a rating of 7 points or better. Solutions due to Justin Foster are available at http://www.math.lsu.edu/~mcgehee/2090/Exe-L2s.pdf

On Monday I'll make some remarks about the test and then continue talking about topics in linear algebra.

Preparing for Test One on September 29. In general, you should expect test questions from the reading, the problems assigned to do on your own, the Exercises, and/or the lectures. Be able to look at a DE, determine what method will work, and carry out the method.

• You should know what a direction field is, and how to draw the direction field of a given DE. Given a picture of the direction field of a DE, you should be able to deduce at least some information about how solutions behave.
• You should understand and be able to use the separation of variables method.
• You should understand and be able to use the integrating factor method for solving equations of linear type.
• You should recognize and understand logistic equations and know how to solve them by the separation of variables method.
• You should understand and be able to use various methods for second-order linear constant-coefficient DEs, homogeneous and non-homogeneous.
• You should understand the Wronskian and its applications.
• You should be able to solve a homogeneous system of two second-order linear DEs in two unknown functions, such as arises in the two-mass, two-spring mechanical vibrating system.
• You should be familiar with the terminology for DEs, including the terms that link the mathematical model with vibrating systems, for example: forced, unforced, damped, undamped, underdamped, overdamped, critically damped, beats, resonance, period, frequency, quasiperiod.

A copy of Test 1 is at http://www.math.lsu.edu/~mcgehee/2090/T1.pdf. A key is at http://www.math.lsu.edu/~mcgehee/2090/T1Key.pdf.

Reading and exercises to do on your own:

• Section 3.3: 1, 3, 5, 11.
• Section 3.4: 9, 11, 13, 15, 19, 21. Read the notes on problems 16-31.
• Section 3.5: 1, 3, 7, 9, 11, 17.
• Section 3.6: 1, 3, 5, 7, 9, 19, 23, 25.
• Section 4.1: 1, 2, 3, 5, 13.
• Section 4.3, particularly Theorems 4.3.2 and 4.3.3. Do 1, 3, 5, 7, 23, 25.

Diagonal matrices. The effect of multiplying a matrix on the left, on the right, by a diagonal matrix. The exponential of a diagonal matrix. Examples of first-order vector-valued DEs and their solution by the eigenvector-eigenvalue method. Defective, nondefective matrices. The case of complex eigenvalues and eigenvectors.

Matrix methods will be developed soon for solving the two-mass, two-spring problem. For an indication of how the problem will be set up in matrix notation, see http://www.math.lsu.edu/~mcgehee/2090/TwoMassesTwoSprings.pdf.
 

Systems of linear equations in two or three unknowns: A Summary. The matrix of coefficients in such a system may be associated with a linear mapping, and we may identify its kernel (null space) and its range. If the matrix is m-by-n, then the kernel's dimension (also called the nullity) and the range's dimension (which equals the rank of the matrix) add up to n.

Suppose that the matrix of coefficients is a square matrix, say n by n. In that case, the determinant is defined. If it equals zero, then the matrix is singular. If it is nonzero, then the matrix is nonsingular, and the solution to the system of equations always exists and is unique (in other words, the nullity is 0 and the rank is n..

In any event, once the augmented matrix is put into row echelon form, one can fairly easily say for which right-hand side vectors a solution exists and, when a solution does exist, describe the set of all solutions. In the nonsingular case, one may proceed to put the matrix into reduced row echelon form, and obtain the inverse of the matrix of coefficients if that is desired.

Interpreting linear maps on finite-dimensional Euclidean spaces geometrically. Rotations. Pure magnifications. Eigenvalue, eigenvector, eigenspace, eigensystem, defective and non-defective matrices, systems of linear equations, phase portraits.

The exponential of a matrix is perhaps the deepest and most powerful mathematics you've yet seen in this course. You're familiar with the exponential function of a real variable, and now we define the exponential of a matrix. It turns out that for the problem x' = A x, a fundamental matrix can be obtained by finding the exponential of the matrix At. In fact, that's the transition matrix at t=0.
 

Lecture topics: Examples, including cases with non-defective and defective matrices, complex eigenvalues, and a two-mass, two-spring system. Diagonalization (for non-defective matrices), trace, determinant, fundamental matrix, transition matrix, initial value problems, orbital portraits.

Reading and exercises to do on your own:

• Section 6.5: 1, 3, 5, 7, 11, 13, 19.
• Section 6.6: 1, 3, 5, 7, 17, 19, 21, 23.
• Section 6.7: 1, 5, 7.
• Section 6.9: "Summary of Results."
• Section 8.1. (We will not use the methods of Section 8.2.)
• Section 8.3: 1, 3, 7, 9, 13.
• Section 8.4.
• Section 8.5: 1, 3, 5.

Problem Set 4, due Wednesday, October 13, is at http://www.math.lsu.edu/~mcgehee/2090/Exe-PS4.pdf Do at least 3 of the 8 problems; you may do up to 8. At the same URL, as of October 18, you will find a solution set; the solutions are not all in order. They include work by students in the class: Christian Naquin, Justin Foster, and Abitha Murugeshu.

For a one-page summary of how to find a fundamental matrix, go to http://www.math.lsu.edu/~mcgehee/2090/FM.pdf.

Lecture topics: Examples with complex eigenvalues. The method of variation of parameters for a nonhomogeneous problem. Introduction to Laplace Transforms and their applications. The table of transforms; deriving its entries from the definition; redundancy in the table.

Problem Set 5, due Wednesday, October 20, is at http://www.math.lsu.edu/~mcgehee/2090/Exe-PS5.pdf Do at least 3 of the problems; you may do all of them if you like. Some are for double and triple credit. After October 20, solutions will be posted at that same URL.

Lecture topics: Laplace Transforms.

Test 2, Wednesday, October 27, will cover the work of Weeks 7, 8, and 9; and Problem Sets 4 and 5. Test questions may represent anything that appears in suggested reading and problems, the lectures, or the Exercises. Start with matrices. You should try to understand matrix multiplication thoroughly. Can you find two square matrices A and B such that AB and BA are not the same? Given a matrix C , let D be C but with its first two rows exchanged; can you find a matrix E such that EC = D ? There's a collection of sample problems at http://www.math.lsu.edu/~mcgehee/2090/SamplesT2.pdf.

Test 2 is available at http://www.math.lsu.edu/~mcgehee/2090/T2.pdf. and a key to Test 2 at http://www.math.lsu.edu/~mcgehee/2090/T2Key.pdf.
 

Problem Set 6, Due Wednesday, November 10 is at http://www.math.lsu.edu/~mcgehee/2090/Exe-PS6.pdf Do at least 3 of the problems; you may do all of them if you like. One is triple credit. After November 10, solutions will be posted at that same URL.

Lecture topics: More on Laplace transforms. Readings and problems to do on your own:

• Section 9.1: 1, 5, 9 (--those are to be doing using the definition), 13, 15, 21 (--the double-angle formula will help), 31, 33, 35 (--I did this in class).
• Section 9.2: All the odds, 7-21 (develop your technique, using the table, and at the same time try to learn at least some of the entries). Feel free to use a computer to do the partial-fractions decomposition. But you should know how to do it by hand.
• Section 9.3: 1, 3, 9.
• Section 9.4: 1, 7, 13, 25, 27, 29.
• Section 9.5: 1, 9, 11, 15, 17, 21, 25, 27, 29, 33, 39, 43, 47, 53 (--do each problem thoughtfully; develop your technique and familiarity with the table).
• Section 9.6: 1, 3, 5, 7, 9, 11.
• Section 9.7: 1, 3, 5, 7, 9, 13, 17, 23, 27, 29, 35.
• Section 9.8: 1, 3, 5, 11, 15.
• Section 9.9: 1, 3, 9, 11, 15, 19.

Objectives: To learn the definition of the Laplace transform, and become efficient at finding the transform by using the definition. To learn how to find the Laplace transform of a function by using a table. To learn how to find inverse Laplace transforms by using a table. To become proficient in the techniques. Although you are using a table, you should try to commit more and more of the table to memory.
 

Lecture Topics: More on Laplace transforms. Solving first-order linear systems, x' = A x + f by the Laplace Transform method.
 

Test 3 is on Wednesday. It will cover Laplace transforms and their applications, not including vector-valued equations. Here's a suggested procedure for studying.

• Know the definition of the Laplace transform of a function.
• As an exercise, prepare your own table of Laplace transforms, referring to notes as little as possible. Know how to use the definition of the Laplace transform to obtain the entries.
• Know how, using the table and your experience from practice, to obtain the inverse Laplace transform in as many cases as possible.
• Pay special attention to the Dirac Delta function and the Unit Step function.
• Pay special attention to the convolution, and its use in solving Volterra equations.
• Practice solving differential equations using the Laplace Transform Method.

A copy of Test 3 and a key are available at http://www.math.lsu.edu/~mcgehee/2090/T3Key.pdf.
 

The lectures will be a combination of review for the final exam and some new material. Topics: Phase portaits. Equations of Cauchy-Euler type, changed into constant-coefficient equations by means of a change of variable. Solving nonhomogenous equations by the method of undetermined coefficients. A mixing problem (Exercise 9 on p. 500) that leads to a 2-dimensional linear nonhomogeneous DE--setting it up, solving by eigenvalue/eigenvector methods, solving by Laplace Transforms.

Read Section 8.11 and do the odd-numbered problems 1-13.

Problem Set 7, due Wednesday, November 24, is at http://www.math.lsu.edu/~mcgehee/2090/Exe-PS7.pdf. No late papers will be accepted. There are three problems, all related to one 2-mass 2-spring system with damping. A solution set will be posted at the same URL after November 24, including some work by Bob Nystrom.

The Final Exam:You should study thoroughly the material represented on the three hour exams, using the advice and guidelines given above in this file. Further advice and review will be provided in the lectures during the last few lectures. Notes: (1) It is likely that a defective matrix and/or a matrix with complex eigenvalues will appear on the final. (2) A complete Laplace transform table will be provided. (3) Study the phase portrait problems on Problem Set 8.

Problem Set 8, due Wednesday, December 1 is at http://www.math.lsu.edu/~mcgehee/2090/Exe-PS8.pdf. You may do as many of the problems as you like. No late papers will be accepted.

The lectures will continue to be part review, part new material. Topics: First order equations. Three types: Exact. Linear. Separable. The use of slope fields. Re-read and review the Chapter 1 assignments of Week 1. In addition, read Section 1.9, pages 71-77 up to "Integrating Factors" and do the odd-numbered problems 1-13.