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General Information

Course ID (CB01A and CB01B)
MATHD002A
Course Title (CB02)
Differential Equations
Course Credit Status
Credit - Degree Applicable
Effective Term
Fall 2023
Course Description
Topics in the course include methods of solving ordinary differential equations and selected applications.
Faculty Requirements
Course Family
Not Applicable

Course Justification

This course satisfies the CSU GE/Breadth requirement Area B4 Mathematics and Quantitative Reasoning. This course satisfies the IGETC requirement Area 2 Mathematical Concepts and Quantitative Reasoning. This course is a required core course for the AS-T degree in Mathematics. This course emphasizes the solution and analysis of differential equations.

Foothill Equivalency

Does the course have a Foothill equivalent?
Yes
Foothill Course ID
MATH F002A

Course Philosophy

Formerly Statement

Course Development Options

Basic Skill Status (CB08)
Course is not a basic skills course.
Grade Options
  • Letter Grade
  • Pass/No Pass
Repeat Limit
0

Transferability & Gen. Ed. Options

Transferability
Transferable to both UC and CSU
CSU GEArea(s)StatusDetails
CGB4CSU GE Area B4 - Mathematics/Quantitative ReasoningApproved
IGETCArea(s)StatusDetails
IG2XIGETC Area 2 - Mathematical Concepts and Quantitative ReasoningApproved
C-IDArea(s)StatusDetails
MATHMathematicsApprovedC-ID MATH 240

Units and Hours

Summary

Minimum Credit Units
5.0
Maximum Credit Units
5.0

Weekly Student Hours

TypeIn ClassOut of Class
Lecture Hours5.010.0
Laboratory Hours0.00.0

Course Student Hours

Course Duration (Weeks)
12.0
Hours per unit divisor
36.0
Course In-Class (Contact) Hours
Lecture
60.0
Laboratory
0.0
Total
60.0
Course Out-of-Class Hours
Lecture
120.0
Laboratory
0.0
NA
0.0
Total
120.0

Prerequisite(s)

MATH D001D or MATH D01DH (with a grade of C or better)

Corequisite(s)

Advisory(ies)

ESL D272. and ESL D273., or ESL D472. and ESL D473., or eligibility for EWRT D001A or EWRT D01AH or ESL D005.

Limitation(s) on Enrollment

(Not open to students with credit in the Honors Program related course.)

Entrance Skill(s)

General Course Statement(s)

(See general education pages for the requirements this course meets.)

Methods of Instruction

Methods of instructions may include but not limited to:

Lecture and visual aids

Discussion and problem solving performed in class

Collaborative learning and small group exercises

Collaborative projects

Use of various technologies including graphing utilities and computer software

Assignments

  1. Required reading from the text, case histories(optional)
  2. Problem-solving assignments
  3. Projects (optional) may include those involving the use of technology

Methods of Evaluation

  1. Periodic quizzes and/or assignments from sources related to the topics listed in the curriculum are evaluated for completion and accuracy in order to assess students’ comprehension and ability to communicate orally and in writing of course content. The students shall receive timely feedback on their progress.
  2. Projects (optional) - Projects may be used to enhance the student's understanding of topics studied in the course in a group or individual formats where communicating their understanding orally through classroom presentation or in writing. The evaluation is to be based on completion and comprehension of course content. The students shall receive timely feedback on their progress.
  3. At least three one-hour exams without projects or at least two one-hour exams with projects are required. In these evaluations, the student is expected to provide complete and accurate solutions to differential equation problems that include both theory and application by integrating methods and techniques studied in the course. The student shall receive timely feedback on their progress.
  4. One two-hour comprehensive final examination in which the student is expected to display comprehension of course content and be able to choose methods and techniques appropriate to the various classes of differential equation problems and models. The student shall have access to the final exam for review with the instructor for a period determined by college and departmental rules.

Essential Student Materials/Essential College Facilities

Essential Student Materials: 
  • None.
Essential College Facilities:
  • None.

Examples of Primary Texts and References

AuthorTitlePublisherDate/EditionISBN
Zill, DennisA First Course in Differential Equations with Modeling ApplicationsBrooks/Cole, Thomson Learning Publishing Co.11th edition, 2018

Examples of Supporting Texts and References

AuthorTitlePublisher
Differential Equations
An Introduction to Differential Equations Order and Chaos
Fundamentals of Differential Equations and Boundary Value Problems
Fundamentals of Differential Equations
Elementary Differential Equations with Boundary Value Problems

Learning Outcomes and Objectives

Course Objectives

  • Explore the development and classification of differential equations
  • Construct differential equation models from social and natural sciences and engineering.
  • Apply analytical, qualitative and numerical methods to solve first order differential equations including the existence and uniqueness theorem in the development of the methods and solutions.
  • Apply analytical methods to solve second and higher order linear differential equations and some special nonlinear equations and include the existence and uniqueness theorems in the development of the methods and solutions.
  • Solve systems of Linear ordinary differential equations with constant coefficients.
  • Find power series solutions to linear ordinary differential equations with variable coefficients, initial value problems
  • Use Laplace transforms to solve ordinary linear differential equations with constant coefficients, initial value problems.

CSLOs

  • Construct and evaluate differential equation models to solve application problems.

  • Classify, solve and analyze differential equation problems by applying appropriate techniques and theory.

Outline

  1. Explore the development and classification of differential equations
    1. Classify differential equations by
      1. type
      2. order
      3. linearity
    2. Investigate historical development of differential equations in mathematics and science through various human activities
  2. Construct differential equation models from social and natural sciences and engineering.
    1. Set up and examine first-order linear models such as but not limited to
      1. Mixture models
      2. Exponential growth/decay models
      3. Suspended cable models
      4. Newton's law of cooling models
      5. Supply/Demand models
      6. Electrical networks
    2. Set up and examine first-order nonlinear models such as but not limited to
      1. Spread of disease models
      2. Competition models
      3. Chemical reaction models
    3. Set up and examine higher order linear and nonlinear models such as but not limited to
      1. Deflection of beams
      2. Buckling of beams
      3. Embedded beams
      4. Spring mass systems
      5. Electrical networks
      6. Variable mass
  3. Apply analytical, qualitative, and numerical methods to solve first-order differential equations including the existence and uniqueness theorem in the development of the methods and solutions.
    1. Study the existence and uniqueness theorem for first-order linear differential equations
      1. Find and investigate solutions and intervals of definition
      2. Investigate singular solutions
    2. Solve first order homogeneous and non-homogeneous initial value problems including
      1. Separable equations
      2. Linear equations
      3. Exact equations
      4. Non-exact equations
      5. Non-linear equations
        1. Equations with homogeneous coefficients
        2. Bernoulli equations
    3. Apply qualitative methods to investigate first-order differential equations
      1. Direction/Slope fields
      2. Phase lines and phase planes
    4. Apply numerical methods to solve first-order differential equations
      1. Numerical integration
      2. Single-step methods
        1. The Euler method
        2. The Runge-Kutta method (optional)
  4. Apply analytical methods to solve second and higher-order linear differential equations and some special nonlinear equations and include the existence and uniqueness theorems in the development of the methods and solutions.
    1. Study the existence and uniqueness theorem for higher-order linear differential equations
      1. Find and investigate solutions and intervals of definition
      2. Investigate singular solutions
      3. Investigate boundary value problems
    2. Solve homogeneous linear ordinary differential equations
      1. Definitions and terminology
        1. The Wronskian, dependence, and independence
        2. Fundamental sets
        3. The superposition principle
        4. Reduction of order
      2. Solve linear differential equations with constant coefficients
      3. Solve Cauchy-Euler equations
    3. Solve non-homogeneous linear ordinary differential equations with constant coefficients
      1. Find particular solutions using the method of undetermined coefficients
        1. Superposition approach
        2. Annihilator approach (Optional)
      2. Find general solutions
    4. Solve non-homogeneous linear ordinary differential equations with variable coefficients
      1. Find particular solutions using the method of variation of parameters
      2. Apply the superposition principle to find general solutions
    5. Solve nonlinear higher-order ordinary differential equations using
      1. Reduction of order
      2. Substitution methods
      3. Approximation methods, at least one of the following
        1. Taylor Polynomial of degree n
        2. Use of Numerical Solvers
    6. Solve application problems from Engineering and Science including but not limited to
      1. Vibrating Springs
      2. Bending and deflections of beams
      3. Non-linear Springs
      4. Electric circuits
      5. Applications including Harmonic Oscillators
  5. Solve systems of linear ordinary differential equations with constant coefficients.
    1. Solve first-order systems using
      1. Substitution method
      2. Elimination method
    2. Solve higher-order systems using
      1. Substitution method
      2. Elimination method
    3. Explore solutions to first-order differential equations and 2x2 systems using direction fields, phase lines, and phase planes.
  6. Find power series solutions to linear ordinary differential equations with variable coefficients, initial value problems
    1. Study the existence and uniqueness theorems for power series solutions to equations with polynomial coefficients
    2. Find power series solutions about
      1. Ordinary points
      2. Regular singular points
  7. Use Laplace transforms to solve ordinary linear differential equations with constant coefficients, initial value problems.
    1. Compute Laplace transforms for various functions
    2. Understand properties of Laplace transforms
      1. Linearity
      2. Operational Properties
        1. Translation of axes
        2. Derivatives of transforms
        3. Transforms of periodic functions
        4. Convolutions
        5. Transforms of impulse functions (optional)
    3. Find inverse Laplace transforms
    4. Compute Laplace transforms of derivatives
    5. Use Laplace transforms to solve Linear differential equations with constant coefficients, initial value problems that include
      1. Continuous inputs
      2. Piecewise continuous inputs
      3. Piecewise periodic inputs
      4. Unit impulse function inputs (optional)
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