# MAT 311 Differential Equations

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### Course Description

Topics covered include first and second order linear equations including existence and uniqueness theorems, series solutions; nonlinear equations; systems of linear equations. Other topics may include the Laplace transform, qualitative theory.

3 units credit.

### Prerequisites

MAT 211 and MAT 271 with grades C or better.

### Text

Texts are chosen by the instructor. For example:

Notes on Diffy Qs by Jiří Lebl. Electronic text available for free at http://www.jirka.org/diffyqs/ or order a paperback copy.

### Course Requirements, Tentative Schedule of Class Meetings and Topics, Readings, Assignments and Due dates, Exams

A schedule of class meetings, topics, assignments, due dates, exam dates, etc. will be provided by instructor. See your class syllabus.

Here is an example course, based on the above text.

• 1st Week
• Chapter 0. Introduction. 0.2 Introduction to differential equations. Types: ordinary, partial. What solutions are. General and particular solutions. Examples. Introduction to software for solving ODEs, plotting slope fields, etc.
• Chapter 1. First order ODEs. 1.1 Integrals as solutions.
• 1.2 Slope fields.
• 2nd Week
• 1.3 Separable equations.
• 1.4 First order linear equations. Integrating factor, or variation of parameters.
• 3rd Week
• 1.5 Substitution.
• 1.7 Numerical methods: Euler, Runge Kutta.
• 4th Week
• 1.6 Autonomous equations.
• Some philosophy regarding deterministic systems, Newton's law $$f=ma$$ and 2nd order ODE's with initial conditions.
• Test
• 5th Week
• Example of Picard Iteration
• More about the Existence and Uniqueness Theorem. Picard iteration. Existence and uniqueness for systems of ODEs (alias vector-valued ODEs), and its relation to second and higher order ODEs. Example where existence and uniqueness theorem does not apply: bucket of water with a hole.
• Chapter 2 Higher order linear ODEs. 2.1 Second order linear ODEs. Superposition, existence and uniqueness.
• 2.2 Constant coefficient 2nd order linear odes.
• 6th Week
• 2.3 Higher order linear ODEs.
• 2.4 Mechanical vibrations.
• 2.5 Nonhomogeneous 2nd order linear equations.
• 7th Week Oct. 5-9
• 2.4 Mechanical vibrations (cont.)
• Solving pendulum equation numerically with software
• 8th Week
• 2.6 Forced oscillations and resonance. How to solve $$y''+ay'+by=f(x)$$ when $$f(x)\neq 0$$.
• Hearing beats (using software and audio channel).
• Chapter 3 Systems of linear ODEs. 3.1 Introduction.
• Phase portrait and trajectory
• 9th Week
• 3.2 Matrices and linear systems.
• 3.3 Linear systems of ODEs.
• 10th Week
• 3.4 Eigenvalue method.
• Matrix algebra and eigenvalues on a computer
• 3.5 Two dimensional systems and vector fields.
• 11th Week
• Test
• Chapter 4 Fourier Series and PDEs. 4.1 Boundary value problems.
• 12th Week
• 4.1 Boundary value problems (cont).
• 4.2 Trigonometric series.
• (skip 4.3-4.5).
• Fourier series for sawtooth function (example, with software).
• 13th Week
• 4.6 PDEs, separation of variables, and the Heat Equation.
• Derivation of the heat equation.
• 14th Week
• Derivation of 1 dimensional wave equation.
• 4.8 D'Alembert solution of 1 dimensional Wave equation
• Animated solution of wave equation (using software).
• 15th Week
• Derivation of 2 dimensional heat equation $$u_{xx}+u_{yy}=u_t$$.
• 4.9 Steady State Temperature and the Laplacian $$u_{xx}+u_{yy}=0$$
• Example: Laplace's equation on a rectangle.

The final exam is given at the date and time announced in the Schedule of Classes.

### Learning Objectives

After completing MAT 311 the student will

• find the solution for linear and nonlinear first order differential equations using the techniques of separable equations, exact equations, and integrating factors
• find the solution for second order linear differential equations with constant coefficients
• find particular solutions to nonhomogeneous second order equations using the methods of undetermined coefficients and variation of parameters
• find series solutions to second order linear equations with polynomial coefficients near an ordinary point, and near a regular singular point
• solve homogeneous and nonhomogeneous first order linear systems with constant coefficients
• determine existence and uniqueness of solutions of differential equations and systems of equations
• use the above techniques to analyze and solve applied problems.

### Computers and Calculators, Computer Literacy

Most instructors encourage the use of machines, calculators computers, phones etc., for analyzing data. The use of machines may be restricted during examinations or at certain other times. Ask your instructor for the policy in your class.

Students are not expected to be programmers or to know any particular computer language before starting this class. Some instructors may expect students to be able to access information on the internet, or to use calculators, or to learn to use particular software with instruction. Basic skill in algebra and the use of mathematical symbols, order of operations etc., and the willingness to read and follow instruction manuals and help files will suffice.

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in this course. The instructor determines the relative weights of these factors and the grading scale. See the syllabus for your particular class.

### Location of Class Meetings

Classes meet on the dates and room announced in the official Schedule of Classes. This is a traditional, face-to-face class.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates, Make-Up Work, Missed Exams, and Extra-Credit Assignments

Due dates and policy regarding make-up work and missed exams are set by the instructor. Instructors may, or may not, choose to offer extra credit assignments. If extra credit assignments are offered they will be available to all students.

The mathematics department does not tolerate cheating. Students who have questions or concerns about academic integrity should ask their professors or the counselors in the Student Development Office, or refer to the University Catalog for more information. (Look in the index under "academic integrity".)

### Accomodations for Students with Disabilities

Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Disabled Student Services (DSS) and to talk with me about how we best can help you. All disclosures of disabilities will be kept strictly confidential. Please note: you must register with DSS to arrange an no accommodation. For information call (310) 243-3660 or send an email message to dss@csudh.edu or visit the DSS website http://www4.csudh.edu/dss/contact-us/index or visit their office WH D-180

### Behavioral Expectations

We all are adults so behavior rarely is an issue. Just follow the Golden Rule: "do unto others as you would have them do unto you" then everything will be fine.

The university must maintain a classroom environment that is suitable for learning, so anyone who insists on disrupting that environment will be expelled from the class.

Revision history:

Prepared by J. Barab 2/4/00. Revised 4/28/01, 7/25/06, 1/10/15 (G. Jennings).