Blog on Edwards and Penney
Differential Equations and Boundary Value Problems:
Computing and Modeling, 3rd edition

Page numbers refer to the 3rd edition, and might have changed in the current edition.

Chapter 1 - First-Order Differential Equations

• 1.1 - Differential Equations and Mathematical Models
• 1.2 - Integrals as General and Particular Solutions
  • Page 11: The remark distinguishing "a" general solution from "the" general solution is useless clutter - it will interest or help only the most logically-inclined student. It would be more helpful to show an example where a general solution method fails to find some particular solution because of some hidden assumption in the general method.
  • The section is theoretically weak. In class, I like to show the students that even if they cannot find an antiderivative of f explicitly, the differential equation dy/dx=f(x) can still be solved by writing down y=y₀+Int^x_x0 f(u) du and recalling the Fundamental Theorem of Calculus (the part of it that most students don't remember). You have to emphasize that the definite integral is a number (the signed area under the graph of f), so that this formula gives an actual value for f(x), for each x. Thus you really do have (in principle) a function y=y(x) that solves the differential equation. The problem is that the book uses the traditional lousy "indefinite integral" notation for antiderivatives, which confuses the matter because just writing down y=Int f(x) dx + C does not give you any actual values for the solution y, and thus does not solve the problem in the same sense that the definite integral formula does.
  • The "velocity and acceleration" problems and methods should be familiar already to students from calculus (or physics) courses.
  • Overall, this seems a rather weak section with which to begin the course.
• 1.3 - Slope Fields and Solution Curves
  • I use the Iode software to help teach about direction fields, and so I de-emphasize the sketching of direction fields by hand. In particular, I ignore the topic of "isoclines" completely.
  • The Existence and Uniqueness Theorem is certainly worth discussing in class, although for those students of a practical mind-set, it will be "clear" already from the direction field that a solution should exist for any reasonable first-order differential equation. The fact that a solution might only exist locally, and might (for example) blow up in finite time, is certainly important. But the other pathologies that the text treats, such as the equation x(dy/dx)=2y which has infinitely many solution curves through the origin, can be skipped without harm.
  • [Aldo Manfroi] The Existence and Uniqueness Theorem has been simplified in the 3rd edition, by assuming throughout that f and its y-derivative are continuous (whereas in the 2nd edition, only the uniqueness statement assumed that the y-derivative of f is continuous). Probably this makes life simpler for the students, but exercises 11-20 need to be updated accordingly, and no longer make sense as stated, since the hypotheses are now the same for the existence and uniqueness parts of the theorem.
• 1.4 - Separable Equations and Applications
  • I tell my students to write the y-antiderivative of 1/y as ln(y), not ln(|y|). The fussiness of using the absolute value is useless in practice (any counterexamples?!). And anyway, from a more advanced viewpoint one can always use the complex logarithm and add an imaginary constant of integration (i Pi), instead of using the absolute value inside the logarithm; note that after exponentiation, this constant becomes a factor of -1. So in my experience, there is no need to bother with the absolute value in the logarithm.
• 1.5 - Linear First-Order Equations
  • The steps on page 47 should start with
    Step 0: rearrange the differential equation into the standard form: y'+P(x)y=Q(x). See the First Order Linear handout.
  • The method would be easier for students to understand if the author wrote "antidifferentiate" instead of writing "integrate".
  • A good example to treat is x'(t)+cx(t)=a cos(kt) + b sin(kt), which is a first-order analogue of the kind of second-order equation treated extensively in Chapter 3. The solution consists of two parts: an exponentially decaying response to the initial condition, and an oscillatory response to the periodic forcing.
  • I like to do one of the mixture problems, to show students another example of how to arrive at a differential equation modeling a real world situation. (They are weak at that skill.)
• 1.6 - Substitution Methods and Exact Equations
  • I omit Exact Equations, because I've never seen them arise naturally except from "conservation of energy" (in which case the level surfaces of the energy already give implicit solutions to the differential equation). Why is this topic in the book? Just for historical reasons, or are there some important applications I don't know about? If there are no applications that are really important, then the author should say so, and relegate the method to the Exercises.
  • Due to lack of time, I also omit "reducible" equations where the independent variable is missing. It is a cute reduction, but doesn't seem to arise too often (although a professor at Courant did once ask me, in conversation, how to solve such an equation, which had arisen in his research).
  • A couple of points that the text ought to make but does not are: if the DE involves yy' then the substitution v=y² might be useful (since then v'=2yy'), and similarly if the DE involves (cos y)y' then the substitution v=sin y might be useful (since then v'=(cos y)y'), and so on. Students should observe closely the form of the differential equation, and look for useful patterns!

Chapter 2 - Mathematical Models and Numerical Methods

• 2.1 - Population Models
  • This section is interesting enough, but I skip it and instead work some of the "population" ideas into my treatment of Section 2.2.
• 2.2 - Equilibrium Solutions and Stability
  • Phase diagrams should always be drawn vertically, not horizontally as in the textbook. Reason: students find it much easier to understand the direction of the arrows on a phase diagram if they can imagine it as the vertical axis for a sketch of solution curves (with an "up" arrow in the phase diagram corresponding to solution curves that are increasing, and similarly for "down" arrows). For example, look at Figure 2.2.1 (solution curves) and Figure 2.2.2 (phase diagram), and imagine how much easier it would be to explain these figures to students if you could put the phase diagram vertically alongside the sketch of solution curves.
  • [Aldo Manfroi] The way the diagrams are drawn, it makes it look like the equilibrium solutions are reached in a finite amount of time - so I always tell the students not to draw different curves touching each other.
  • The definition of stability on page 92 is correct, but this formal epsilon-delta definition of the general concept of stability is precisely the wrong thing to state in this class. Every example we encounter actually satisfies the more intuitive notion of asymptotic stability (see page 372). In fact I see no harm at this level in simplifying a little and just defining asymptotic stability to mean that if the solution x(t) ever gets close to the value c, then it must tend to c as t -> infinity. The graphical examples make the meaning perfectly clear. (If I were teaching a course to math majors, I would be more precise in my definition.)
  • The treatment of bifurcation that culminates in the bifurcation diagram (Figure 2.2.12) at the end of the section is very nice, if you have time.
  • A good in-class example is to sketch the graph of a function f(x) that crosses the x-axis a few times (that is, for which f(x)=0 has a few roots), and then ask the students to draw the phase line for the differential equation dx/dt=f(x).
• 2.3 - Acceleration-Velocity Models
  • This section is not worth covering on its own, although it is a handy reference for the air resistance problems. I cover the "resistance proportional to velocity" problem when I cover Section 1.4, and then I challenge students to investigate the "resistance proportional to the square of velocity" problem on their own, for falling objects (note the constant of proportionality would have the opposite sign for a rising object).
• 2.4 - Numerical Approximation: Euler's Method
  • I take students to the computer lab and have them work through Iode Lab 2, rather than lecturing on Euler's method. Then I follow up in class with a discussion of error behavior (which is of order h, so that halving the step size cuts the error by a factor of 2), and examples where the method does not work well e.g. highly oscillatory direction fields, or solutions that blow up in finite time.
  • The text material is good, although Example 2 on the dropping baseball is too artificial to be helpful.
  • A good in-class exercise is to have the students compute the first few iterations of the Euler method by hand, for some simple example.
• 2.5 - A Closer Look at the Euler Method
  • The Improved Euler Method is covered in Iode Project 2. I don't cover this material in class. The text ought to emphasize more strongly that the error is of order h^2, so that halving the step size cuts the error by a factor of 4 (in general).
• 2.6 - The Runge-Kutta Method
  • There is no time to cover this, so I just point interested students to the textbook.

Chapter 3 - Linear Equations of Higher Order

• 3.1 - Introduction: Second-Order Linear Equations
  • The text is too enamoured of the general theory, and should instead start by emphasizing the fundamental examples y''+k²y=0 and y''-k²y=0, showing how to solve them using sin/cos and exponentials. Then remind students about the definition and properties of sinh/cosh, including values at the origin, derivatives, hyperbolic Pythagoras identity cosh²-sinh²=1, and their graphs; maybe mention the connection to hyperbolas. Do NOT assume your students already know the hyperbolic trig functions. Then show how y''-k²y=0 can also be solved using sinh/cosh.
  • These same examples should be used to illustrate the existence and uniqueness theorem; point out to students that the initial conditions work out much more nicely for sinh/cosh than for the exponentials.
  • The superposition principle for linear homogeneous equations should be immediately followed by the corresponding principle for linear nonhomogeneous equations. This shows students that adding two solutions of a DE need not give a solution of the same DE.
  • Wronskians are a needless distraction to students in a "methods"-oriented course, and should not be mentioned at all. That is, we should skip over Theorem 3 (Wronskians of Solutions) and go straight to Theorem 4 (General Solutions of Homogeneous Equations). It is Theorem 4 that we really care about, and we only use Theorem 3 to prove Theorem 4. So Theorem 3 should be relegated to the Appendices.
  • As a further warning to anyone thinking of proving these theorems, notice that the proof of General Solutions Theorem 4 depends on the uniqueness part of the Existence and Uniqueness Theorem 2, which is proved only in the appendix (and indeed the uniqueness part is only sketched there). So even if you go to the trouble of teaching your students about Wronskians, they still will not get the complete theoretical picture until you have proved the Existence and Uniqueness Theorem. Given this rather complicated chain of dependences, I can't see why the author thinks it is so crucial to introduce Wronskians. If I wrote the book, I'd relegate Wronskians to the exercises.
  • Note that in practice, we never compute a Wronskian in order to check linear independence. It is one of the travesties of the traditional ODE course that students end up believing they should compute some meaningless (to them) determinant in order to see if a collection of functions is independent. In practice, if you have two functions then you simply check whether or not one is a multiple of the other; and if you have more than two functions arising as solutions of an ODE, then they will almost certainly have some structure that allows you to see linear independence quite readily. For example, when studying constant coefficient, linear homogeneous ODEs, we find exponential solutions for which linear independence is easily proved directly e.g. to show that {e^x, e^2x,e^3x} are linear independent, just assume they are linearly dependent and deduce a contradiction (using the differing growth rates at infinity).
  • When I teach linear systems, I will say a bit more about Wronskians; but not much more.
  • If I were teaching math majors, then I'd certainly teach the full theory of about linear independence and Wronskians. But my target audience consists of engineers, not math majors, and Wronskians seem a waste of time for engineers. (Counterexamples?)
  • For the case of "repeated roots" for constant coefficient, linear homogeneous ODEs, I like to show the students where the "x" comes from in the solution xe^rx. For example, consider two distinct roots r and s, so that (e^sx-e^rx)/(s-r) solves the ODE; letting s tend to r yields xe^rx. Thus if s=r then we would expect xe^rx to be a solution of the ODE. This is not a proof, but it is close enough. It also gives me an opportunity to talk about why roots are "generically" distinct, and why we should regard the case of equal roots as a limiting case of the distinct root case.
  • I treat here also the case of complex roots for a second order linear constant coefficient homogeneous equation. (Strangely, the textbook delays treatment of complex roots until Section 3.3.)
  • I definitely spend half an hour discussing complex numbers with my students before doing the case of complex roots. Reason: an astounding number of students are either ignorant of complex numbers, or afraid of them, or are deeply suspicious of their validity. I stress that "there is nothing imaginary about complex numbers!" Here's a good way to do it. Define a complex number to be a pair (a,b) of real numbers, and define the multiplication rule for complex numbers to be: (a,b)(c,d)=(ac-bd,ad+bc). Then for notational simplicity I write (a,b)=a+bi, where the "i" is just a symbol that indicates that "a" is the first real number in the pair and "b" is the second number. Then you observe (0,1)=i. And the multiplication rule implies i²=(0,1)(0,1)=(-1,0)= -1+0i. That is, i²=-1.
    The deeper point here is that by enlarging our idea of what is a "number" (in fact, by considering pairs of real numbers, which we call complex numbers) we have found a "number" whose square is -1.
    Then I explain that multiplication by i corresponds to rotation by 90 degrees: i(a+bi)=-b+ai, and I draw a picture to illustrate. In particular, i i = i²=-1 just says that rotating the point (0,1) by 90 degrees gives the point (-1,0).
    After these foundational remarks, I define the complex exponential using the Taylor series, and assert that all the usual algebraic rules like e^z+w=e^ze^w still hold. Then I check Euler's formula e^it = cos(t) + i sin(t) by using the Taylor series of the three functions. Then you can express polar coordinates in complex form as a+bi = re^it where r=(a²+b²)^1/2 is the magnitude of the complex number a+bi, equivalently of the vector (a,b).
    Finally, I advertize Math 446 (or Math 448) as a good course for learning more about the wonderful properties of complex numbers.
    Note: This treatment of complex numbers got some unusually positive reaction from the better students. They really seemed to appreciate having the complex theory demystified.
• 3.2 - General Solutions of Linear Equations
  • Sometimes I don't cover this material until after Section 3.4, because Sections 3.3 and 3.4 are mostly about the second order case anyway.
  • This section should only be treated briefly, because it is so similar to the second order case in Section 3.1. I give a handout and put it up as an overhead, and talk the students through it.
  • Just like in the second order case above, I omit Wronskians. "Down with Wronskians!"
  • The new concept in this section is linear independence for more than two functions. Students find this tricky to understand, and it is worth spending time on the definition, and examples. Actually, when I'm teaching Math 385, which does not cover systems of ODEs, I just define linear independence to mean that no one of the functions can be written as a linear combination of the others. This definition works well in practice, even though it is not the most elegant definition, because students can easily remember and check it.
    But when I teach Math 386, which includes systems, because the linear algebra is more important I go the extra step and show how the above definition of linear independence is equivalent to the usual one about the only linear combination that equals zero being the trivial combination.
  • If you still don't believe me about Wronskians, then try the following with your class. Have half the class check linear independence of {e^x,e^2x,e^3x} using the Wronskian and the other half by checking that no one of these functions can be written as a linear combination of the other two functions. See which half of the class has a more satifactory understanding of the meaning of linear independence.
  • One should not get too carried away with the higher-order material in Sections 3.2 and 3.3, because the most-used differential operators in the world are second order. Of course, there are some physical situations that yield higher order operators (e.g. vibrating beams give a fourth order operator), but students should first thoroughly understand the second order case and all its ramifications e.g. mechanical vibrations in the next section.
    One might argue that higher order equations are important because they motivate the study of systems of first order equations. But systems of first order equations arise perfectly naturally all by themselves, and don't need motivational help from artificially complicated higher order equations.
  • It is a deficiency of the textbook that it fails to advise the reader properly, with comments like my "don't get carried away with the higher-order material in Sections 3.2 and 3.3, because the most-used differential operators in the world are second order".
    Not every topic is equally important! The author ought to guide the reader better by offering pungent opinions. (Or does the publisher prefer bland inoffensiveness?!)
• 3.3 - Homogeneous Equations with Constant Coefficients
  • The "operator" notation for differential equations is a good conceptual step in this section, and the idea that you can factor a differential operator is important. But this topic can wait till after Section 3.4 Mechanical Vibrations. I think it's more important to get students quickly to some applications.
• 3.4 - Mechanical Vibrations
  • There are three good real-world examples: the mass-spring system, the RLC electrical circuit (which should be taken from Section 3.7 and inserted here, with a very brief statement of the analogies m=L, c=R, k=1/C), and the pendulum (which is nonlinear, but becomes approximately linear for small oscillations). Incidentally, the period of a pendulum is not independent of the amplitude when you use the full nonlinear equation - independence only holds after you make the linearizing approximation.
  • Students have a surprising amount of difficulty with converting solutions from the form Acos(wt)+Bsin(wt) to the form Ccos(wt-g), particularly if the phase shift g is in the second or third quadrant (so that one must add pi to arctan(B/A)).
  • You might want to tell your class what a dashpot is.
• 3.5 - Nonhomogeneous Equations and Undetermined Coefficients
  • Rule 2 on page 202 is good, but why on earth does the author confuse matters with the different-looking Rule 1 on page 198?! It is much better to state Rule 1 exactly the same way as Rule 2 except without the factor of x^s that is needed in Rule 2 to handle the duplication. Then students only have to remember one coherent approach, not the mishmash of different approaches given in the book.
  • See my handout on undetermined coefficients, where I state Rule 1 and Rule 2 in a consistent fashion and give examples for students to work in class.
  • See also my handout on variation of parameters, where I state the method and give examples for students to work in class. (In class, I also give a proof that the method works.)pwd
• 3.6 - Forced Oscillations and Resonance
  • The students need lots of practice to help them understand and distinguish the different phenomena (beating, resonance, practical resonance). And of course some get confused by amplitude response graphs, because they are not used to thinking of the forcing frequency as a variable.
  • Overall the textbook does a good job, although I think it would help to make up a summary diagram encapsulating the material of Section 3.4 (free motion) and Section 3.6 (forced motion). This diagram could show the four cases (free, undamped; free, damped; forced, undamped; forced, damped), along with the behavior of the complementary and particular solution in each case, and brief reminders of the qualitative behavior associated with that case (e.g. beating, practical resonance).
• 3.7 - Electrical Circuits
  • I don't cover this section. Instead I just briefly mention the electrical-mechanical analogy while teaching Section 3.4. I think Section 3.7 should be moved to some kind of "appendix" at the end of the chapter, or into some kind of "application" section. Because there really is no new mathematics developed here.
• 3.8 - Endpoint Problems and Eigenvalues
  • I defer this till after Chapters 4 and 5, since it does not seem to be needed until Chapter 9.
  • Page 231 is the natural place to state the Fredholm Alternative; but the author does not, curiously.
  • What is lacking here is any mention of Orthogonality of Eigenfunctions, which of course is the crucial fact later for Fourier series in Chapter 9. I work through a handout with my class, to show that eigenfunctions that satisfy the same type of BC and have different eigenvalues are automatically orthogonal. Orthogonality is then deduced for the standard trigonometric system.

Chapter 4 - Introduction to Systems of Differential Equations

• 4.1 - First-Order Systems and Applications
  • When it comes to phase portraits, I think the book needs to do more to help students understand the connection between the the x- and y-solution curves and the phase portrait and the underlying physical situation. For example, if you analyze a simple mass-spring system, then you want students to be able to look at the circular trajectory in the phase plane and identify which points of the trajectory correspond to maximum rightwards displacement of the mass, which points correspond to the mass passing through the equilibrium position while at maximum speed, and so on.
  • The basic pedagogical principle here is that when you introduce a new method of representing information (such as a phase portrait), you ought to help students make mental connections between this new representation and the old representation (such as x- and y-solution plots).
• 4.2 - The Method of Elimination
  • Not covered.
• 4.3 - Numerical Methods for Systems
  • Not covered.

Chapter 5 - Linear Systems of Differential Equations

• 5.1 - Matrices and Linear Systems
  • In-class worksheet
  • The treatment of determinants is deficient because it fails to say what determinants mean! (I ascribe this to the undue influence of algebraists on the teaching of linear algebra...) The determinant should be defined as the (signed) volume of the parallelepiped spanned by the column vectors of the matrix (or the row vectors). Then in order to compute this obviously useful quantity, one develops the usual determinant formulas. Since time is short in this class, I only justify the determinant formula in 2 dimensions: draw the parallelogram with edges [a c]^T and [b d]^T and compute the area: you find it is ad-bc, which is therefore our definition of determinant for a 2x2 matrix [a b \\ c d].
  • The whole section is poorly structured. The first part consists of a rapid-fire survey of basic matrix algebra. (This should be a separate section.) Then it transitions to a somewhat rambling restatement of material on superposition and general solutions, complementary and particular solutions, all of which we are familiar with from the scalar case in Chapter 3. The text then ought to (but does not) say the following. "So just like in Chapter 3, we face three separate issues. 1. Finding n linearly independent complementary solutions of the homogeneous equation. 2. Finding a particular solution of the nonhomogeneous equation. 3. Finding the constants c₁,...,c_n so that the initial conditions are satisfied." Instead of saying this and then laying out a plan for dealing with all three issues, the text just launches into dealing with issue 3 (satisfying the initial conditions), showing how to solve the system of linear equations by row reduction.
  • Row reduction summary
  • See my comments on Wronskians in Section 3.1 above. The same comments apply here.
• 5.2 - The Eigenvalue Method for Homogeneous Systems
  • The examples are good, but the overall feeling of the section is "I'll tell you how to find a solution, but I won't ask you to think about the phase portrait". There should really be some clear statement, in this section, of what the phase portraits look like in the standard 2 dimensional examples: two positive real roots (source), two negative real roots (sink), one positive and one negative (saddle), a complex pair of roots with positive real part (spiral source), a complex pair of roots with negative real part (spriral sink). One can do all this in the easiest possible canonical cases, and just say "the general cases look like distorted versions of these phase portraits" - this statement about distortion is readily understood if you do just one example where the eigenvectors are not in the coordinate directions, such as Example 1 in the text. An excellent source for this (standard) material is Hirsch, Smale and Devaney "Differential Equations, Dynamical Systems, and an Introduction to Chaos".
  • The section curiously fails to mention that if a matrix is real and symmetric, then all the eigenvalues are real. (This fact is useful in Section 5.3, for example.)
  • Another odd omission is the failure to prove that distinct eigenvalues have linearly independent eigenvectors. This very important fact is more-or-less stated on page 302, but without proof. Why is this not proved, when the text investigates less-fundamental ideas in gory detail (e.g. Section 5.4)?! The proof is not long, and could easily be done here.
• 5.3 - Second-Order Systems and Mechanical Applications
  • This is a very nice section, with a lovely interplay of theory and physical intuition.
  • In Example 2, the assumption that the buffer springs disengage when stretched should not be mentioned until the end, where it is used.
  • The final subsection "Periodic and Transient Solutions" is not much use, since damping terms are not considered anywhere else in the section. And note formula (39) needs to be rephrased as x(t)-x_p(t) -> 0, in order to make sense.
  • The author could make mass-spring systems seem more appealing by showing how the wave equation for compressional waves can arise as a limit of mass-spring systems as the number of oscillators goes to infinity. Examples: sound waves, seismic P-waves.
• 5.4 - Multiple Eigenvalue Solutions
  • This section seems like overkill. The theory is too deep for the students to understand (if the students have not seen linear algebra before), and the applications are not sufficiently compelling to make the reader believe that the work involved is worthwhile.
  • I go lightly: first show a simple example of a deficient eigenvalue, then explain the general algorithm on page 337, and then apply the algorithm to the example.
• 5.5 - Matrix Exponentials and Linear Systems
  • I ignore the material on the matrix differential equation X'=AX. The students have enough to think about already with the vector equation x'=Ax, and I don't want to burden them further. So I skip the first 4 pages of the section, the fundamental matrix etc.
  • Instead I begin with the following motivation: the familiar equation x'=ax with initial condition x(0)=x₀ has solution x(t)=e^atx₀, and so by analogy, we expect the vector equation x'=Ax with initial condition x(0)=x₀ to have solution x(t)=e^Atx₀. But clearly this means we need to find a satisfactory definition of the exponential of a matrix.
  • After definining the matrix exponential by the Taylor series, I show e⁰=I, and compute the exponential of a diagonal matrix. Then I consider the examples A=[0 1 // 0 0] and B=[0 0 // 1 0], and directly compute e^At and e^Bt and e^(A+B)t (the last one involves cosh and sinh). This provides dramatic proof that the familiar laws of exponentials can fail for matrices, since e^At e^Bt does not equal e^(A+B)t here.
  • Then I prove the theorem that if A and B commute then equality does hold. It's not hard, and I don't understand why the book relegates this beautiful result to the Problems. The key point of the proof is that the binomial expansion depends on commutativity e.g. (A+B)² = A²+AB+BA+B² = A²+2AB+B² provided AB=BA.
  • Another curious omission is diagonalization, which provides both a method for computing the exponential of a matrix, and enables one to see the connection between the solution x(t)=e^Atx₀ and the solution in terms of eigenvectors and eigenvalues that we found in Section 5.2.
  • At the end of the section, I show the derivative of e^At is Ae^At, by using the Taylor series, so that our original motivation is correct: the vector equation x'=Ax with initial condition x(0)=x₀ does have solution x(t)=e^Atx₀.
• 5.6 - Nonhomogeneous Linear Systems
  • I don't like the content or organization of this section. The section fails to adequately explain the parallels with the scalar case, and it does not treat the second order vector case, and worst of all, in my view, it fails to use the fundamental idea of decomposing the forcing term and the solution into eigenvector series. The eigenvector decomposition idea is fundamental in differential equations, and was used already in Section 3.8 and will be used again in Chapter 9 (partial differential equations). Surely it should be used here too!
  • Here is what I cover in class, instead.
    1. First order linear constant coefficient nonhomogeneous. (a) Explain the integrating factor method (the integrating factor is e^-At). This works when A is a constant matrix, and not in general when A depends on t (because the derivative of e^B(t) need not equal e^B(t)B'(t), when B(t) and B'(t) do not commute). (b) Explain eigenvector decomposition: decompose the forcing term into a linear combination of eigenvectors, with variable coefficients f_j(t), and guess that the solution can be written as a linear combination of eigenvectors, with variable coefficients g_j(t). Substitute this guess into into the differential equation to obtain a scalar first order equation for each g_j. Solve that scalar equation by the usual integrating factor method.
    2. First order linear variable coefficient nonhomogeneous. Here I follow the nice treatment in the text.
    3. Second order linear constant coefficient nonhomogeneous. (a) Explain Undetermined Coefficients very briefly: it is like in the scalar case (Section 3.5) except with vector coefficients. And note the slight difference in the case of duplication, noted on page 360. (b) Explain eigenvector decomposition. Like in the first order case above, except solve for g_j using scalar Undetermined Coefficients (Section 3.5).
    4. Second order linear variable coefficient nonhomogeneous. Mention Variation of Parameters. Omitted.
  • Then give them lots of practice at these methods!

Chapter 9 - Fourier Series Methods

• 9.1 - Periodic Functions and Trigonometric Series
  • Fourier series are presented in a rather low-level way. For instance, the trigonometric orthogonality formulas on page 574 are proved using trig identities, with no mention of the fact that orthogonality is to be expected because the sines and cosines are eigenfunctions of y''+lambda y=0 satisfying periodic boundary conditions. (See my comments above on Section 3.8.)
  • Also, there is no description of the vector analogy, where we expand a vector v=ai+bj+ck in terms of orthogonal vectors i,j,k, and find the coefficients a,b,c by taking dot products with i,j,k and using orthogonality. I do this example before even writing down the Fourier series formula, because then students can see the analogy in the Fourier formula, as we develop it.
• 9.2 - General Fourier Series and Convergence
  • Regarding the Convergence Theorem, I wish the book emphasized the importance of sketching the graph over at least two full periods, so that any jumps at the "endpoints" become clearly visible.
• 9.3 - Fourier Sine and Cosine Series
  • I point out in class that a sine series can be obtained either by odd extension and a full Fourier series, or by simply using that the sine functions on the original interval are orthogonal (because they are eigenfunctions for Dirichlet boundary conditions). This dispels the feeling that sine series are just some kind of trick.
  • The material on Termwise Differentiation of Fourier Series fails to clearly state the main point: we want to differentiate the series of x(t), and x(t) is smooth because it solves a differential equation. Thus termwise differentiation is valid. [The discussion in the textbook of counterexamples to term-by-term differentiation is irrelevant to our needs.]
• 9.4 - Applications of Fourier Series
  • After teaching this section, I give students a summary of the methods in Sections 9.3 and 9.4 for solving ODEs by Fourier series. Particularly, I explain how to detect and deal with resonance.
• 9.5 - Heat Conduction and Separation of Variables
  • Consider the brief comment on page 613: "the series solution...usually converges quite rapidly, unless t is quite small, because of the presence of the negative exponential factors." This is correct, but it fails to emphasize the main point in a form students can readily understand. The comment should say that "one finds in practice that after a short time t, only the first one or two terms in the solution u(x,t) need be added up, because the other terms will be neglibly small. In other words, the high modes (terms with large n) decay extremely fast and do not contribute much to the solution, in practice."
    This comment should be accompanied by snapshops of a solution that is very wiggly at t=0, but which has become basically an arch of a sine curve by time t=1.
    Unfortunately, the book has a lot of this kind of writing - insights are presented using declarative language rather than action-oriented or algorithmic language. The reader is left wondering what he or she is actually supposed to do with these statements presented by the author.
• 9.6 - Vibrating Strings and the One-Dimensional Wave Equation
  • The D'Alembert solution is derived by trig identities from a series solution on the interval 0 < x < L. Aaaaagggghhhhh!!!!!!! Please derive it instead by changing variable (to characteristic coordinates) in the wave equation on the whole line -Infty < x < Infty, in the standard way, to get u=F(x+ct)+G(x-ct). Then do some examples with F=G (initial velocity zero) and with F=-G (initial displacement zero).
• 9.7 - Steady-State Temperature and Laplace's Equation
  • No comment.