| Credit- Degree applicable | | Effective Quarter: Fall 2022 | I. Catalog Information
| MATH 2A | Differential Equations | 5 Unit(s) |
| | (See general education pages for the requirement this course meets.) (Not open to students with credit in MATH 2AH.)
Prerequisite: MATH 1D or MATH 1DH (with a grade of C or better).
Advisory: EWRT 211 and READ 211, or ESL 272 and 273. Lec Hrs: 60.00
Out of Class Hrs: 120.00
Total Student Learning Hrs: 180.00 Topics in the course include methods of solving ordinary differential equations and selected applications. |
| Student Learning Outcome Statements (SLO)
| | | Construct and evaluate differential equation models to solve application problems. |
| | | Classify, solve and analyze differential equation problems by applying appropriate techniques and theory. |
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II. Course Objectives | A. | Explore the development and classification of differential equations |
| B. | Construct differential equation models from social and natural sciences and engineering. |
| C. | 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. |
| D. | 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. |
| E. | Solve systems of Linear ordinary differential equations with constant coefficients. |
| F. | Find power series solutions to linear ordinary differential equations with variable coefficients, initial value problems |
| G. | Use Laplace transforms to solve ordinary linear differential equations with constant coefficients, initial value problems. |
III. Essential Student Materials IV. Essential College Facilities V. Expanded Description: Content and Form | A. | Explore the development and classification of differential equations |
| 1. | Classify differential equations by |
| 2. | Investigate historical development of differential equations in mathematics and science through various human activities |
| B. | 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 |
| b. | Exponential growth/decay models |
| d. | Newton's law of cooling models |
| 2. | Set up and examine first order nonlinear models such as but not limited to |
| a. | Spread of disease models |
| c. | Chemical reaction models |
| 3. | Set up and examine higher order linear and nonlinear models such as but not limited to |
| C. | 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 |
| a. | Find and investigate solutions and intervals of definition |
| b. | Investigate singular solutions |
| 2. | Solve first order homogeneous and non-homogeneous initial value problems including |
| 1. | Equations with homogeneous coefficients |
| 3. | Apply qualitative methods to investigate first order differential equations |
| b. | Phase lines and phase planes |
| 4. | Apply numerical methods to solve first order differential equations |
| 2. | The Runge-Kutta method (optional) |
| D. | 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 |
| a. | Find and investigate solutions and intervals of definition |
| b. | Investigate singular solutions |
| c. | Investigate boundary value problems |
| 2. | Solve homogeneous linear ordinary differential equations |
| a. | Definitions and terminology |
| 1. | The Wronskian, dependence and independence |
| 3. | The superposition principle |
| b. | Solve linear differential equations with constant coefficients |
| c. | Solve Cauchy-Euler equations |
| 3. | Solve non-homogeneous linear ordinary differential equations with constant coefficients |
| a. | Find particular solutions using method of undetermined coefficients |
| 2. | Annihilator approach (Optional) |
| 4. | Solve non-homogeneous linear ordinary differential equations with variable coefficients |
| a. | Find particular solutions using method of variation of parameters |
| b. | Apply the superposition principle to find general solutions |
| 5. | Solve nonlinear higher order ordinary differential equations using |
| c. | 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 |
| b. | Bending and deflections of beams |
| e. | Applications including Harmonic Oscillators |
| E. | Solve systems of Linear ordinary differential equations with constant coefficients. |
| 1. | Solve first order systems using |
| 2. | Solve higher order systems using |
| 3. | Explore solutions to first order differential equations and 2x2 systems using direction fields, phase lines and phase planes. |
| F. | 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 |
| b. | Regular singular points |
| G. | 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 |
| 2. | Derivatives of transforms |
| 3. | Transforms of periodic functions |
| 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 |
| b. | Piecewise continuous inputs |
| c. | Piecewise periodic inputs |
| d. | Unit impulse function inputs (optional) |
VI. Assignments | A. | Required reading from text, case histories(optional) |
| B. | Problem-solving assignments |
| C. | Projects (optional) may include those involving the use of technology |
VII. 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 |
VIII. Methods of Evaluating Objectives | A. | 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 student’s comprehension and ability to communicate orally and in writing of course content. The students shall receive timely feedback on their progress. |
| B. | Projects (optional)
Projects may be used to enhance the student's understanding of topics studied in the course in group or individual formats where communicating their understanding orally through classroom presentation or in writing. The evaluation to be based on completion and comprehension of course content. The students shall receive timely feed back on their progress. |
| C. | 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 feed back on their progress.
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| D. | 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. |
IX. Texts and Supporting References | A. | Examples of Primary Texts and References |
| 1. | Zill, Dennis. "A First Course in Differential Equations with Modeling Applications, 11th edition." 2018, Brooks/Cole, Thomson Learning Publishing Co. (* The text used most recently in our department.) |
| B. | Examples of Supporting Texts and References |
| 1. | Blanchard, Devaney, and Hall. "Differential Equations." Brooks/Cole Publishing Co., 4th edition, 2011. |
| 2. | Diacu, Florin. "An Introduction to Differential Equations Order and Chaos." W.H. Freeman and Company, 2000-08-15. |
| 3. | Nagle, Kent R., Saff, Edward B. and Snider, Arthur David. "Fundamentals of Differential Equations and Boundary Value Problems, 7td edition." Addison Wesley Longman, 2018. |
| 4. | Nagle, Staff., Saff, and Snider, "Fundamentals of Differential Equations, 9thd edition." Pearson Addison Wesley, 2018 |
| 5. | Elementary Differential Equations with Boundary Value Problems, by William F. Trench, Publishers: Brooks/Cole Thomson Learning, Free Edition 1.01 (December 2013) |
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