Department of Mathematics
University of Wisconsin – Madison

Differential Equations Courses

UW-Madison offers three distinct introductory-level differential equations courses.

Some of the named option plan for the major in mathematics require a course in differential equations. The standard major does not. It may be possible to use a proof-based differential equations course to complete the requirement of a named option plan. Students should consult the Guide for more information. Note that in all versions of the major and certificate, only one of the following courses may be used to fulfill any of the requirements.

The purpose of this page is to describe the essential differences between the three introductory-level differential equations courses.

MATH 319 (Techniques in Ordinary Differential Equations)

MATH 319 is a first course in differential equations. Students typically see differential equations, which are critical to modeling many systems in the sciences and engineering, first in MATH 222, but this course provides a survey of a greater range of types of differential equations and techniques for solving them than is introduced in the calculus sequence. In this course, the emphasis is typically on computational aspects of the subject with some lighter treatment of the more theoretical points.

Students who complete this course and would also like an introduction to linear algebra should consider MATH 340.

In summary, MATH 319…

  • Is ideal for students who need comprehensive functional knowledge of introductory differential equations for applications in the sciences and engineering;
  • Is not by itself sufficient for enrollment in advanced math courses;
  • Covers material in MATH 320 and therefore credit for only one of MATH 319 or 320 can be applied to the math major or certificate;
  • Covers material in MATH 376 and therefore credit for only one of MATH 319 or 376 can be applied to the math major or certificate.

Suggested further courses are…

  • MATH 340 for those interested in the applied analysis sequence;
  • MATH 415 for those interested in seeing related concepts from dynamical systems as applied to a variety of scientific fields; or
  • MATH 421 for those interested in advanced undergraduate math courses above the 500-level.

MATH 320 (Differential Equations and Linear Algebra)

Differential equations and linear algebra are crucial for the analysis and control of many dynamical systems such as electrical networks, mechanical structures, chemical and biological networks. Those applications lead to often very large systems of coupled differential equations. The study of the equilibria, stability and control of such systems requires the concepts and tools of differential equations and linear algebra. MATH 320 is a first course in the study of differential equations and linear algebra. As such, students who complete this course can consider themselves as having some of the content of MATH 319 (Techniques in Ordinary Differential Equations) and some of the content of MATH 340 (Elementary Matrix and Linear Algebra). This course lends itself to the MATH 321-322 applied analysis sequence.

Students who have completed MATH 320 will need to complete one of MATH 421; the applied analysis sequence MATH 321-322; or MATH 467 before moving on to 500-level courses that require introductory proofs coursework.

In summary, MATH 320...

  • Is useful for students interested in classical applications of mathematics (e.g., physics, engineering, continuous modeling, etc.);
  • Covers material in MATH 319 and therefore credit for only one of MATH 319 or 320 can be applied to the math major or certificate;
  • Covers material in MATH 340 and therefore credit for only one of MATH 320 or 340 can be applied to the math major or certificate;
  • Covers material in MATH 341 and therefore credit for only one of MATH 320 or 341 can be applied to the math major or certificate;
  • Covers material in MATH 375 and therefore credit for only one of MATH 320 or 375 can be applied to the math major or certificate;
  • Covers material in MATH 376 and therefore credit for only one of MATH 320 or 376 can be applied to the math major or certificate;
  • Is not by itself sufficient for taking advanced math courses;
  • Is offered with an Accelerated Honors (!) version in some semesters. This version is suggested for students in the AMEP program.

Suggested further courses are...

  • The applied analysis sequence MATH 321-322, which covers more mathematics useful for traditional applications;
  • MATH 415, which includes both continuous and discrete models of changing systems;
  • MATH 421 for an introduction to more formal mathematical arguments; or
  • MATH 443 for further study of linear algebra in an applied context.

MATH 376 (Topics in Multivariable Calculus and Differential Equations)

MATH 376 is an Accelerated Honors (!) course that continues the study begun in MATH 375. This course completes the study of multivariable calculus and then focuses on differential equations. It also provides students with an introduction to proofs and proof-writing. In terms of subsequent coursework, MATH 375 generally fulfills any prerequisite that includes MATH 234 as well as courses requiring introductory background in differential equations.

In summary, MATH 376:

  • Is Honors-level;
  • Enrollment is restricted to those who have completed MATH 375;
  • Is accepted in both the major and certificate programs;
  • Is not a course you can take if you have credit for one or more of MATH 234, 319, 320, 340, or 341;
  • Is good additional practice with proofs and proof-writing;
  • Covers material in MATH 319 and therefore credit for only one of MATH 319 or 376 can be applied to the math major or certificate;
  • Covers material in MATH 320 and therefore credit for only one of MATH 320 or 376 can be applied to the math major or certificate;
  • Will give students access to a number of advanced-level undergraduate math courses.

More information on the MATH 375/376 sequence can be found on our Honors calculus page.