Math 256

COURSE DESCRIPTION: (for catalog)

An introductory course that focuses on the application of ordinary differential equations as mathematical models for a variety of disciplines and the interpretation of ordinary differential equations through analytical, graphical, and numerical techniques. A systems approach is taken with the relevant linear algebra concepts developed as needed. A TI-89 calculator is required and a computer laboratory may be included.

PREREQUISITE:

MTH 253 with a grade of C or better

INSTRUCTIONAL MATERIALS REQUIRED OF STUDENT: (text, supplies, etc.)

LIST THE PERFORMANCE OBJECTIVES OR STUDENT LEARNING COMPETENCIES AS DESCRIBED IN THE INSTRUCTIONAL LEARNING SYSTEM FOR THIS COURSE:

First Order Differential Equations
• Modeling
• Analytic Techniques
• Numerical Techniques
• Qualitative Techniques
• Related Theorems
• Related Vocabulary

Linear Systems
• Modeling
• Analytic Techniques
• Numerical Techniques
• Qualitative Techniques
• Related Theorems
• Related Vocabulary

Numerical Methods
• Associated Errors
• Improved Methods

PERFORMANCE OBJECTIVES

A successful student will need to demonstrate the ability to:

Prerequisites

• Apply and synthesize vocabulary, concepts and techniques from prerequisite courses with those of this course

Vocabulary

• Correctly use mathematical vocabulary associated with ordinary differential equations (ODE)

Modeling

• State the advantages and disadvantages of mathematical models
• Explain the implications of a modeling ODE in terms of the situation being modeled
• State why dp/dt = kp is not a realistic population model and how it could be improved
• Model basic applied situations with ordinary differential equations
• State two "real world" examples that would be appropriately modeled with autonomous ODEs
• State three examples of situations in which an ODE with a parameter would be appropriate model

Verification of Solutions

• Determine whether a proposed solution to an ODE is or is not an actual solution by substitution

Theorems

• Determine if the hypotheses of the existence and uniqueness theorems are satisfied
• Explain the value and limitations of the existence and uniqueness theorems
• Explain the difference between necessary and sufficient conditions
• Prove the Linearity Principle and other basic theorems

Analytic techniques

• Identify ODEs in which separation of variables would be an appropriate technique
• Use separation of variables (showing steps) to solve the appropriate ODEs
• Identify linear ODEs
• Use an integrating factor technique (showing steps) to solve the linear first-order ODEs
• Use a guess and check technique to solve appropriate ODEs
• Apply the Linearity Principle to express the general solutions of ODEs

Qualitative techniques

• Describe what is meant by a qualitative solution
• Sketch slope field and solutions to basic ODEs without the aid of a computer/calculator
• Describe the qualitative solution of an ODE including equilibrium solutions, sinks and sources
• Describe the appropriateness and value of a phase line solution to an ODE
• Determine the phase line associated with autonomous ODEs
• Determine the bifurcation values associated with a given family of autonomous ODEs
• Sketch the bifurcation diagram associated with a given family of autonomous ODEs

Numerical techniques

• Describe how Euler's Method approximates the solution to an ODE
• Apply Euler's method to determine successive approximations to the solution of an initial value ODE
• State the relative advantages and disadvantages of Euler's method
• State how Euler's method can be improved, as well as, the limitations of those "improvements"
• Describe how Improved Euler's and Runga-Kutta Method approximates the solution to an ODE
• Apply Improved Euler's and Runga-Kutta method to determine successive approximations to the solution of an initial value ODEs
• State the relative advantages and disadvantages of Improved Euler's and Runga-Kutta methods
• Describe the effects of finite arithmetic on numerical solutions to ODEs and how they can be minimized

Modeling with Systems of Ordinary Differential Equations

• Explain the implications of a modeling system of ODEs in terms of the situation being modeled
• Model basic applied situations, including predator/prey, mixture, harmonic oscillator problems, with systems of ODEs

Verification of Solutions to Systems of Ordinary Differential Equations

• Determine if a proposed solution to a system of ODEs is or is not an actual solution by substitution

Theorems regarding Systems of Ordinary Differential Equations

• Determine if the hypotheses of the existence and uniqueness theorems for systems of ODEs are satisfied
• Explain the value and limitations of the existence and uniqueness theorems for systems of ODEs
• Prove the Linearity Principle for systems of ODEs and other basic theorems

Analytic Techniques associated with Systems of Ordinary Differential Equations

Qualitative techniques associated with Systems of Ordinary Differential Equations

• Sketch vector fields, direction fields, and slope fields associated with a given ODE or system of ODEs
• Sketch the solution curve in the XY phase plane associated with a graphical solution given in the X(t) and Y(t)-planes for a system of ODEs of the form d(X,Y)/dt = V(X,Y)
• Sketch the graphical solution in the X(t) and Y(t)-planes from a given solution curve in the XY phase plane for a system of ODEs of the form d(X,Y)/dt = V(X,Y)
• Describe the qualitative solution of a system of ODEs including equilibrium solutions, sinks, sources, saddles, centers and separatrixes.
• Describe the fundamental property of the Lorenz system of ODEs
• Describe the importance of the Lorenz system of ODEs and some of the implications
• Construct a table organizing the various possible types of linear systems (sink, spiral sink, source, ), the condition of their eigenvalues, typical phase portraits, and associated solutions

Numerical techniques associated with Systems of Ordinary Differential Equations

• Describe how Euler's Method approximates the solution of a system of ODEs
• Apply Euler's method to determine successive approximations to the solution of an initial value ODE
• Apply Improved Euler's and Runga-Kutta method to determine successive approximations to the solution of a system of initial value ODEs
• State and demonstrate the relative advantages and disadvantages of various numerical methods

Computer/Calculator Competencies

• Sketch vector fields, direction fields, and slope fields associated with a given ODE or system of ODEs using appropriate computer software
• Apply numerical methods to determine approximate solutions, expressed both numerically and graphically, for initial value ODEs (including systems) using appropriate computer software
• Determine the error between an Euler's method solution and a known exact solution using appropriate software
• Determine, when possible, the analytic solution to ODEs using appropriate software
• Transfer, run and modify/edit basic calculator programs

Optional Topic (as time permits)

• The theory and technique of series solutions
• An introduction to Laplace Transforms

GENERAL INSTRUCTIONAL METHODS:

Over the past few years, most calculus courses across the country have been "reformed" and here at MHCC there has been a serious effort to reform the courses prior to calculus as well. So it seems only appropriate that the differential equations class undergo a comparable transformation. A reformed ODE curriculum emphasizes conceptual understanding, mathematical modeling of real-world applications, multiple representations of a problem (and their solutions), appropriate use of available technology, and mathematical problem solving. This represents a shift away from technique mastery and algorithmic skills. For students to see mathematics as an integrated whole, the performance objectives listed on previous pages should be presented in a connected fashion and not treated as discrete topics or concepts as the listing of them in this document might seem to imply.

Calculator: A programmable hand-held calculator is required for this course. Not only can the calculator function as a tool for quick and accurate computation but the opportunity to understand, modify and create programs is one more facet of mathematics education reform. In addition, technology is a component of the real world for which we are preparing our students. We must encourage technological as well as mathematical literacy.

Computer Laboratory: Computer software can be a valuable supplement to the capabilities of a hand-held calculator, exhibiting features and capabilities not available (or at least accessible to the operator) on all of the student's calculators, such as closed-form solutions, graphs of solutions of ODEs, printing, colors, 3-dimensional graphing, etc.

Learning Environment: Students learn in many different ways: by writing, listening, discussing, asking, explaining, reading, etc. The classroom should be structured to support these many learning styles. Thus, teaching should involve a variety of learning opportunities including such methods as:

In addition, out of class work should also include a variety of learning opportunities including:

The classroom should be a cooperative environment managed by the instructor, but focused on the students. Thus, neither a complete lecture format nor a team-based classroom left alone is a successful model. A balance of learning environments, instructor supervision, and student contribution are necessary components of an enriched learning environment. Although the balance of activities in a classroom varies between instructors, a successful learning environment will certainly include a well-balanced mixture of the following essential components:

• Problem solving activities provide the students an opportunity to develop and apply a variety of strategies to solve problems, verify and interpret results with respect to the original problem situation, and generalize solutions and strategies to new problem situations. Through this experience students acquire confidence in using mathematics meaningfully and are able to formulate and solve problems as they exist in the real world and in their field or area of interest.
• Concepts presented in a class should lead to applications in areas such as natural resources, human resources, health services, business and management, industrial and engineering technology, and arts and communication. Through contextual learning, students are able to value the role of mathematics in our culture and ever-changing technological society.
• Guided discovery learning activities must be provided to help the student take responsibility for his/her learning and develop a mechanism to "learn how to learn." By investigating patterns and exploring concrete, pictorial, and graphical models, students create their own understanding of mathematical concepts. Discovery activities also teach students to be adventuresome in their approach to problems - that they need not know the answer before beginning to try something.
• Teams should be constructed to best allow for whole team discussion without any students being left out. Teams should work together in class most days on tasks furthering their understanding of the material and their problem solving/communication abilities. Most discovery activities are completed in teams. (It is recommended that your teams be comprised of no more than five students due to the tables used in our classrooms.)
• Although teams are an extremely important and valuable learning environment, they cannot replace whole class discussions where students share their insights in an interactive lecture with the instructor as the knowledgeable authority. Team activities (especially discovery activities) need to be followed by a discussion/lecture to ensure that all students understand the material.
• To help coordinate these follow-up efforts and to ensure that your class is learning the material, you should be constantly assessing their progress while students are working in teams. Teamwork is not a break for the instructor. You need to be available to answer questions, sometimes guide discussions, facilitate good team behaviors as needed, and gather information about your students' comprehension and ideas.
  This type of assessment will help you select and set up your next learning environment wisely (or adjust the current one as necessary). As students are working on an activity, you may realize that some instructor-led discussion is necessary in the middle rather than just at the end (as you might have planned). You can use your observations to decide whether you should follow a team problem solving session with a class discussion, a lecture, or by having each team put their solution on the board and giving the teams an opportunity to present their approaches. Flexibility is an extremely important part of teaching in a student-centered learning environment.
  Although flexibility is important, you must remember that the course outline must be covered by the end of the term. Sometimes you may need to leave a topic that students are not comfortable with and continue covering material. It is important to spiral back and address these weaker points as the term continues.

EVALUATION CRITERIA FOR STUDENT LEARNING AND MINIMUM ACCEPTABLE LEVEL OF PERFORMANCE

The students enrolled in this course generally have a strong mathematics background coupled with good study skills. But there still will be some students that are weaker than others who require additional attention so as not to be over looked and there will be some that are stronger than others and also require additional attention.

Grades should be based on a balanced variety of grading opportunities spread throughout the term. Although you may not choose to use every method below, a variety of methods give a good perspective of the various facets of a student. Student evaluation must include problems or activities that incorporate and integrate several outcomes, and closely resemble situations that exist in the real world.