# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 22B: Differential Equations

Approved: 2000-09-01 (revised 2024-05-03, Nachtergaele (2011);)
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Elementary Differential Equations and Boundary Value Problems, 11th Edition, by Boyce/DiPrima (Wiley; \$157.12 via Amazon.com)
Search by ISBN on Amazon: 978-1119820512
Prerequisites:
MAT 022A C- or better or MAT 067 C- or better.
Suggested Schedule:
 Lecture(s) Sections Comments/Topics 1 and 2 1.1-3 Introduction and terminology, direction fields, discussion and solution of some ODE 3 2.1 Linear equations; integrating factors 4 2.2 Separable equations 5 2.3-4 Modeling, mechanics; Linear versus non-linear equations 6 2.5 Autonomous equations; Population dynamics 7 2.7 Numerical approximation; Euler’s method 8 2.8 Existence and uniqueness theorem 9 2.9 First order difference equations 10 3.1 Homogeneous 2nd order equations with constant coefficients 11 and 12 3.2-3 Fundamental solutions, linear independence, Wronskian 13 3.4 Complex roots 14 3.5 Repeated roots; Reduction of order 15 3.6 Nonhomogeneous equations; Method of undetermined coefficients 16 3.7 Variation of parameters 17 3.8-9 Applications to oscillating systems 18 6.1 Laplace Transform, definition 19 6.2 Solution of initial value problems with Laplace Transform 20 7.1 Systems of linear ODE, introduction 21 7.2-3 Review of related linear algebra 22 7.4 Basic theory of first order linear systems 23 7.5 Homogeneous linear systems with constant coefficients 24 7.6 Complex eigenvalues 25 7.7 Fundamental matrices 26 7.8 Repeated eigenvalues 27 7.9 Nonhomogeneous linear systems 28 Applications and review

This syllabus is based on 27 50-minute lectures. This usually leaves two lectures for midterms, e.g. midterm one covering the material of Chapters 1 and 2, and midterm two covering the material of Chapters 3 and 6. Alternatively, one can hold one midterm and have a lecture on applications (and/or review) at the end.

There are several interesting options to extend and/or modify this material:

• One could spend additional time in the beginning on examples of different types of ODE, their solution and applications; then summarize the basic methods a bit more quickly.
• Include treatment of linear systems using the Laplace transform.
• Section 2.9 on first Order Difference Equations can be considered optional. Alternatively, it can be expanded with a discussion of iterated maps (e.g. the logistic map).
• Section 2.7 on Numerical Approximations: Euler’s Method could be the starting point for project(s) using Matlab or another software package.
• There are many computations in this course that can be further explored with a mathematical software package such as Matlab, Maple or Mathematica. In particular, Section 2.7 on Numerical Approximations: Eulerâs Method is the starting point for project(s) using Matlab or another software package.
Learning Goals:
Upon successful completion of this course, students will be able to:
• Identify and classify various types of differential equations.
• Solve first-order differential equations using analytical methods, including separation of variables and integrating factor.
• Compute numerical approximations of the solutions to differential equations using Euler’s method.
• Use geometric methods (slope field and phase lines) to qualitatively analyze first order differential equations.
• Determine whether unique solutions are guaranteed to exist.
• Use the characteristic equation to solve second-order linear differential equations with constant coefficients.
• Calculate solutions of second-order linear differential equations by the method of undetermined coefficients and variation of parameters.
• Represent solutions of a second-order linear differential equation in terms of a fundamental set of solutions.
• Use Laplace transforms to solve linear differential equations.
• Solve systems of linear differential equations using matrix methods and interpret the solutions.
• Construct differential equations that model processes in the natural world and solve the equations using appropriate methods.