This document is a summary of Mathematics courses - through differential equations, taken by undergraduate engineering students at some different U.S. schools. The purpose of the summary is to illustrate how other Universities treat the material related to Utah's Math 2250 (Ordinary Differential Equations and Linear Algebra).
The thesis of this web page is that it would behoove the Engineering College and the Mathematics Department at the University of Utah to consider upgrading the current course into one which covers linear algebra and differential equations in more depth and breadth than is possible with the current 3-credit hour, 3 meetings per week format.
The survey begins with a brief look at how other schools in Utah treat Math 2250 or its equivalent.
The bulk of the document is a summary of how some top U.S. Engineering schools treat the equivalent material. These Universities are the top 10 graduate engineering programs in the country, according to recent rankings of of U.S. News and World reports; "U.S. News and World Reports'' spring 99 graduate rankings. They are also rated in the top 11 undergraduate programs among schools granting Ph.D.'s, in the most recent ratings of the same magazine; fall 99 undergraduate rankings.
Utah State teaches Math 2250 as a 4-credit course, meeting for four lectures a week. At Weber State, Math 2250 is also a 4-credit course, although it is apparently not being offered in the 1999-2000 year ( current schedules). Instead, students would have to take 6 total-credit versions of the U's Math 2270-2280 to be taught about linear algebra and differential equations. Utah Valley State College and Southern Utah University do not offer Math 2250; students must also take 6 total-credit versions of the U's Math 2270-2280 to cover the equivalent material at these schools.
At Brigham Young University engineering students who need differential equations or linear algebra take a year-long integrated sequence (6 credits total), called Advanced Engineering Mathematics which covers linear algebra, differential equations, multivariable Calculus, and an introduction to partial differential equations. Of all the non-U Utah schools, B.Y.U. is the only in-state school which possibly teaches the material of Math 2250 with as few hours as does the U, although it does not appear that the 2250 topics are separated from the others in BYU's year-long sequence; there is probably a time savings obtained by integrating the various listed topics into the year-long course, since for example, linear algebra is helpful for all the other topics, and multivarible Calculus and differential equations are helpful for partial differential equations.
Finally, Salt Lake Community College is teaching Math 2250 as a 3-credit offering (to be consistent with the U.), but officially meets 5 times a week in order to cover the material.
The summaries below and the associated links indicate that most schools try to coordinate the teaching of linear algebra and differential equations, although the coordination manifests itself in different ways. Of the 10 schools discussed below, three of them (Stanford, Berkeley, Purdue) cover both topics in one course. Four of the ten have a substantial linear algebra pre-requisite for the DE's course (M.I.T., C.I.T., Georgia Tech., Stanford). Three of the universities teach the linear algebra after the DE's (Michigan, Illinois, Texas). All of the schools except for Carnegie Mellon have the linear algebra and the differential equations linked in some way, if not in the same course then at least in the same sequence or by means of a prerequisite.
Of the 10 Universities outlined below, perhaps all of them use more time than the University of Utah to teach linear algebra and differential equations to their engineering students. The range of semester credit hours alloted to these subjects probably ranges between 4 and 7, with 5 semester credit hours being close to the average. The amount of time which the top engineering schools choose to devote to linear algebra and differential equations may be a reflection of the value which they place on these topics, and of the depth and breadth in which they wish them to be covered.
Of the 8 schools for which textbook information could be found on-line, 5 of them (the schools ranked 1-4 and 6) use the same text as the University of Utah, Edwards-Penney, for differential equations. The other three (ranked 5,7,9) use Boyce and DiPrima. There did not seem to be a pattern in their choice of linear algebra materials.
1-variable Calculus, Math 18.01, is a one semester, 5 credit course (3 lectures plus 2 recitations per week), as is multivariable Calculus, Math 18.02. Math 18.02 includes vector algebra in 3-space, matrices, determinants. Neither 18.01 nor 18.02 syllabi mention differential equations.
Differential Equations, Math 18.03, is a 5 credit course (3 lectures plus 2 recitations/labs per week). Text is Edwards-Penney (same as us), as well as ``Ordinary Differential Equations Using Matlab'' by Polking. Course includes our current differential equations syllabus as well as nonlinear systems, phase plane, and Fourier series solutions to differential equations. The syllabus does not include linear algebra, but has the 18.02 prerequisite in which most of the linear algebra material we teach in 2250 is covered. The course includes computer work, using ``Matlab;'' current course information. A more advanced 4-credit course on linear algebra,Math 18.06, is available for students who are interested (analogous to our Math 2270 or Math 3730).
M.I.T. does not strictly use a ``credit'' system, instead listing courses by ``units'', where units are the sum of class meetings, lab hours, and estimated outside hours per week. Almost all courses are 12 unit courses, for example, Math 18.03 is a (5,0,7) course. One could argue whether a course with 3 lectures and 2 recitations per week would typically be listed as 4 or 5 credits at the U. But note that since the linear algebra which the University of Utah covers in Math 2250 is largely covered in M.I.T.'s prerequisite Math 18.02, the U's 2250 material corresponds to at least 5 semester credit hours at M.I.T.
1-variable Calculus, Math 41-42, is a two quarter 5 ``unit'' course (3 lectures plus 2 recitation sections). The second quarter includes an introduction to differential equations. Sequence can also be taken in 3 quarters, at 4 units each quarter (Math 19-20-21).
Differential Equations is taught as part of an integrated year-long sequence, Math 51-52-53. Each course is 5 quarter-system credits (3 lectures, 2 sections). Math 51 is differential multivariable calculus, with an introduction to linear algebra, Math 52 is multivariable integral calculus, and Math 53 is more linear algebra and differential equations. Math 51 is prerequisite for Math 53. DE's text for Math 53 is Edwards-Penney, topics include ours as well as Chapter 6 on the phase plane. Linear Algebra text is Penney's text ``Linear Algebra''. Course does not appear to have a computer lab component at this time. Counting the linear algebra from Math 51, and converting from quarters to semesters, Stanford's treatment is probably the equivalent of about 5 semester credit hours. current course information
Calculus I, Math 1501, is a 4-credit semester course (3 lectures 2 sections) on single variable calculus. Calculus II, Math 1502, also 4 credits (3+2), includes first and second order linear differential equations, but the bulk of course is linear algebra with topics similar to Math 2270 at Utah. Calculus III, Math 2401, also 4 credits (3+2), is multivariable calculus.
Differential Equations, Math 2403, is a 4 credit course (3 lectures plus 2 recitations/labs per week). The text is Edwards-Penney, chapter 6 on phase plane and non-linear systems is covered, as are chapter 7 on Laplace Transforms, and chapter 8 on power series. The prerequisite is Calculus II, i.e. solid linear algebra foundation in addition to 1-variable calculus. Counting the linear algebra from Math 1502, Georgia Tech probably spends about 6 semester credit hours on material related to the University of Utah's Math 2250.
1-variable Calculus, Math 115-116 is a two-quarter, 4 credit sequence. Math 116 includes an introduction to differential equations, mostly first order with some second order and phase plane material. Multivariable Calculus, Math 215, is a 4-credit (quarter) course.
Differential Equations, Math 216, is a 4 credit (quarter) course, thought of as half of a sequence 216-417, in which the second course is a 3 credit (quarter) course on matrix and linear algebra. Prerequisite is Math 116. Current course information indicates that Edwards-Penney is the text, and that chapters 1-7 are covered, i.e. including chapter 6 on phase plane analysis. There is a computer lab component. For Math 417, Matrix Algebra I, the text is ``Linear Algebra with Applications'', by Bretscher. The combination of a 3-credit and a 4-credit quarter course is roughly equivalent to a 5-credit semester course, and this is about how much time the University of Michigan gives its Engineering students to learn linear algebra and differential equations.
1-variable Calculus, Math 1A-B, is a two semester, 4 credit each, sequence (3 lectures plus 2 sections). Math 1B includes first and second order differential equations, oscillation and damping, power series solutions. Math 53 is a 4-credit course on multivariable Calculus (3 lectures plus 2 sections).
Linear Algebra and Differential Equations, Math 54, is a 4 credit semester course (3 lectures plus 2 recitations per week). Texts are ``Elementary Differential Equations and Boundary Value Problems'' by Boyce and Di-Prima, together with ``Elementary Linear Algebra'' by Hill. Students may take a supplemental 1-unit computer lab with the course, or in a special section (54M) the lab and course are integrated into one four-credit item. The course covers elements of partial differential equations as well as some material similar to the University of Utah's Math 2250. There does not seem to be any on-line material related to this course, beyond the current textbooks.
1-variable Calculus, Math 120-130, is a two semester, sequence, a 5-credit course followed by a 3-credit course. There is no mention of differential equations or linear algebra in the topic lists. Multivarible Calculus, Math 242, is a 3-credit course.
Differential Equations and Linear Algebra are not integrated. A 2-credit course, Math 225, Introduction to Matrix Theory, has Math 120 as prerequisite. A 3-credit course, Math 285, called Differential Equations and Orthogonal Functions, covers ODE's with an optional introduction to P.D.E.s. This course has Math 242 as prerequisite. Different sections of the course cover different options, i.e. systems of differential equations versus partial differential equations. Both tracks cover Fourier series. Text for both tracks is Edwards-Penney. Chapter 6 is not covered. Neither Chapter 7 (Laplace transforms) nor Chapter 8 (power series solutions) are covered in either track. The combination of differential equations and linear algebra (matrix theory) requires a total of 5 semester credit hours at the University of Illinois. For more information on the differential equations course go to the current course home page.
1-variable Calculus and multi-variable Calculus are part of the first-year track, called Probability and Calculus of One and Several Varibles, Math 1abc. Each quarter course is 9 ``units'', where a M.I.T.-like system of counting seems to be in force, and this is a (4,0,5) course. Probably 4 (quarter) credits in the U's accounting system. Basic linear algebra is part of this sequence too.
Differential Equations and Linear Algebra are part of the 2nd-year, 2-quarter sequence Math 2ab, Linear Algebra, Statistics, and Differential Equations. These are also (4,0,5) courses. Differential Equations are covered in 2b, with textbook Boyce and DiPrima. There does not appear to be a current home page, since this course is not taught in the first term. The linear algebra is a little less than half of Math 2a, see this fall's home page.
1-variable Calculus is contained in the year-long sequence Math 21-121, 21-122 with a syllabus very much like the U's, and with first order and constant coefficient second order DE's covered at the end of 122. The course is taught in a 3-lecture 2-section format (called 10 units), probably a 4 credit course. Each course can be further broken down into two minicourses. Multivariable Calculus, Math 21-259, is a 3-lecture 2-section course, (called 9 units).
Differential Equations, Math 21-260, is a 3-lecture 1-section 9 unit course (3-4 credits?) It includes Fourier Series and an introduction to partial differential equations, but is not integrated with linear algebra. As at the U., the prerequisite is the first-year Calculus. There is a separate 3-lecture 0-section course called 21-241, Matrix Algebra, which covers elementary linear algebra ideas. The combination of Math 260 and Math 241 is 6-7 semester credit hours which covers material close to that of the University of Utah's Math 2250 and Math 3150 (5 total credit hours)
1-variable Calculus can be taken in the 2-semester, 4-credit sequence Calculus and Analytic Geometry, Math 173-174, or in the 2-semester, 3-credit sequence Calculus for Technology, Math 221-222. Multivariable Calculus, Math 261, is a 4-credit course.
Differential Equations and Linear Algebra can be taken in an integrated or non-integrated fasion. Math 262, Linear Algebra and Differential Equations covers elements of both in a 4-credit course; Math 266, Ordinary Differential Equations, just covers DE's. The text is Boyce and DiPrima and there is a computer lab component, using Matlab. current home page . A 3-credit course, Math 265, Linear Algebra, is also available. Comparable numbers of people seem to take each of Math 262 and Math 266, judging from the current class schedule. The differential equations courses all have the multivariable course Math 261 as prerequisite. There is a junior level, 3-credit course Math 303, Differential Equations for Engineers, with 262 as a prerequisite, which covers systems of DE's, Fourier series, and an introduction to P.D.E.'s.
1-variable Calculus can be taken is a 1-semester 4 or 5 credit course (3 lecture 2 section), Math 408C. Sequences, series, and beginning multivariable Calculus are covered in Math 408D, also a 3 lecture 2 section course.
Differential Equations, matrices, and multivariable Calculus are part of a year-long sequence Math 427K-427L, Advanced Calculus for Applications. Each course is a 3-lecture 2-section course. Differential Equations are covered in 427K, the matrix theory and multivariable Calculus in 427L. There does not seem to be much web-based information on these courses.