Course of Study Mth 279, Introduction to Differential Equations

Distance Learning Option

Course Title, Number and Description, Required Prerequisite Knowledge

Introduces ordinary differential equations. Includes first order differential equations, second and higher order ordinary differential equations with application. Designed for mathematical, physical, and engineering science programs. Prerequisite: MTH 174 or equivalent.
Lecture 4 hours per week.
4 credits

Required Prerequisite Knowledge:  To succeed in this course a student must have good mastery of Precalculus and first-year differential and integral calculus.

The nature of the course

This course is offered via the Internet and via distributed DVD's in an asynchronous mode. The student will receive instructional information and assignments via these modes and will respond to assignments by submitting work through web forms.

The student must have standard access to the Internet and must have the ability to access the content on the DVD's.  The material on the DVD's is accessible using a variety of media players (e.g., Windows Media Player). 

The instructor is available via web forms (to which students will be introduced at the very beginning of the course), and will normally respond by the end of the day following your submission (and more typically on the same day) with answers to properly posed questions, feedback on your efforts, and other information. Exceptions may occur in the event of Internet problems or other technical events. 

Broad goals and Purpose of the Course

The student will gain proficiency with first-order differential equations, second-order differential equations, systems of linear differential equations, solution of differential equations using Laplace Transforms, and use of these skills in the context of both mathematical and real-world applications.  Goals include but are not limited to:

Specific objectives

Each assigned task and problem constitutes a specific objective, which is to complete that problem or task and understand as fully as possible its relationship to the stated goals of the assignment and to other concepts, problems and situations encountered in the course.

Unless you are the 1 person in 1000 with the combination of aptitude and learning style capable of doing it otherwise, solving problems is the only way to learn the material in this course.

Ability to perform the tasks listed below provide a good foundation for solving the assigned problems, the process of which leads to mastery.

 

Table of Specific Objectives for Mth 279, Introduction to Differential Equations

Module 1:  Introduction; First-Order Linear Equations; First-Order Nonlinear Equations

Assignments 1 - 13

Test #1 covers material through Assignment 13, and is due within a week after completing Assignment 13.

1.      Given a differential equation identify intervals over which solutions can be guaranteed to exist.

2.      In terms of the behavior of the direction field of a given equation discuss on which intervals solutions can be expected to exist.

1.      Given a first-order linear homogeneous equation and an initial condition find and verify its solution.

2.      Solve applications problems by writing and solving appropriate first-order differential equations and interpreting the solutions.

1.       Given a first-order linear homogeneous equation and an initial condition find and verify its solution.

2.      Solve applications problems by writing and solving appropriate first-order differential equations and interpreting the solutions.

3.      Use differential equations to solve basic mixing problems.

1.      Apply appropriate techniques to scale direction fields when the scales of the two coordinate axes differ.

2.      Solve problems involving temperature relaxation, in which the rate of change of the temperature of an object is proportional to the difference between its temperature and that of its surroundings.

3.      Identify and solve separable differential equations with initial conditions.

4.      Identify situations in which are characterized by a constant (positive or negative) growth rate; set up and solve equations to model such situations.

1.      Describe the possible effects on solution curves of a vertical asymptote in a direction field.

2.      Verify that if the total differential of a function of two variables is set equal to a constant, the result is a first-order differential equation.

3.      Test a given differential equation for exactness.

4.      Given a differential equation verified to be exact, solve it.

5.      Given a mixing problem which can be modeled with a first-order differential equation, write the equation and solve it.

1.      Match graphs of solution curves with equations.

2.      Set up and solve first-order equations for the velocity of constant mass under the influence of a constant force in addition to a force proportional to the first or second power of the velocity.

3.      Use the transformation v = y^m to transform a Bernoulli equation into a linear differential equation, solve the equation and verify your solution.

4.      Solve the logistic equation by separation of variables and partial fractions.

1.      Given a situation involving logistic growth, set up and solve the appropriate equation and interpret the solutions.

2.      Given a situation in which the net force on a constant mass is a function of position use the chain rule dv / dt = dv/dx * dx/dt to change the dependent variable from t to x; solve and interpret the resulting equation.

1.      Apply the Euler method to obtain an approximate solution curve for a given first-order differential equation, and place appropriate limits on the accuracy your solution.

2.      Determine whether a result obtained from the Euler method is an overestimate or an underestimate, and place appropriate limits on the extent of the error.

3.      Discuss the effect of step size on the accuracy of an Euler approximation.

4.      Solve Bernoulli equations and interpret the solutions.

1.      Find solutions of second-order homogeneous equations a y ‘’ + b y + c = 0 with constant coefficients by substituting y = e^(r t).

2.      Know the Euler identity and apply it to solutions in cases where r values are imaginary or complex.

3.      Evaluate the Wronskian of set of functions to determine whether it is linearly independent.

4.      Describe the span of a given set of functions.

1.      Determine the intervals on which continuous solutions to a given second-order equation with nonconstant coefficients would be expected to exist.

2.      Show what is obtained if y = u e^(r t) is substituted into a homogeneous linear equation, where r is a solution to the characteristic equation.

1.      Solve second-order linear constant coefficient equations for which the characteristic equation yields a repeated root.

2.      Solve a situation involving an LRC circuit for the case where a voltage source is not present.

3.  Analyze a situation involving motion of an object subject to the force of a spring in which a drag force proportional to velocity is also present.

1.      Solve nonhomogeneous second-order linear equations with constant coefficients using the method of undetermined coefficients.

2.      Solve nonhomogeneous second-order linear equations with constant coefficients using the method of variation of parameters.

3.      Apply Euler’s method to second-order equations.

Module 2:  Second-Order Equations

Assignments 14 - 23

Test #2 covers material through Assignment 23, and is due within a week after completing Assignment 13.

#14

qa 13
  • interpret same eqn as damped pend, LRC

4.1

Section 4.1 Problems

Query 12
  • floating cylinder
  • intervals of solution
  • nature of solution

110307

 

1.      Interpret a given second-order equation with constant coefficients in terms of the stored charge in an LRC circuit, and in terms of the motion of a damped pendulum.

2.  Write the differential equation for a cylinder of given dimensions and density bobbing in a fluid of given density.

3.  Given a second-order linear differential equation with initial condition, determine the maximum interval over which a solution can be guaranteed to exist.

4.  Given a second-order differential equation and the slope at a given point in the plane, determine the concavity of the solution curve in the vicinity of that point.

1.  Solve second-order linear equations with constant coefficients, for which the coefficient of the first derivative is zero.

2.  Verify whether a given set of solutions constitutes a fundamental set on a given interval.

 

1.  Express a given system of linear differential equations as a matrix equation.

2.  Calculate the derivative and/or an antiderivative of a given matrix.

3.  Solve the eigenvalue problem for a given matrix.

4.  Verify whether a given set of solutions to a differential equation is linearly independent.

 

1.  Write the system of equations for and describe the qualitative behavior of the rabbits-and-wolves model.

2.  Write and solve the system of linear equations for a two-tank mixing problem with constant flows.

3.  Solve second-order constant coefficient equations.

4.  Solve problems involving mathematical and real-world applications of second-order constant coefficient equations.

 

 

1.  Analyze the phase plane to describe the general behavior of solutions of the rabbits-and-wolves model.

2.  Write and solve second-order constant coefficient equations to model two-spring systems, RC circuits and LRC circuits.

3.  For various functions f(t) integrate f(t) * e^(-2 t) from t = 0 to infinity.

4.  Solve second-order differential equations for which the characteristic equation has repeated real roots or complex conjugate roots.

5.  Apply the method of reduction of order to find a second solution of the second-order linear equation given one solution.

 

 

1.  Express a third- or higher-order system of linear differential equations with constant coefficients as a system of linear differential equations.

2.  Given a physical system which undergoes linearly damped or free unforced harmonic motion, write the second-order equation (wht appropriate initial conditions) which models its motion, solve the equation, and interpret the solution.

 

 

1.  Using the definition find the Laplace Transform of a given function.

2.  Using the definition express the Laplace Transform of the first derivative of a given function in terms of the Laplace transform of the function itself and an initial condition.

3.  Show that the Laplace Transform is a linear operator.

4.  Correctly identify the specific Laplace Transforms of a function whose transform you have previously derived, and correctly identify the specific function associated with the given transform.

5.  Apply the method of undetermined coefficients, where it is possible to do so, to find a particular solution of a given second-order nonhomogeneous equation, and combine with the solution of the homogeneous equation to obtain a general solution.

 

 

1.  Apply Laplace Transforms to solve the equation y ' = - k y.

2.  Using the definition of Laplace Transform, find the transform of the Heaviside function h(t), and of h(t - alpha).

3.  Use the Heaviside function to construct a square-wave function, and find its Laplace Transform.

4.  Use the definition of the Laplace Transform to find the transform of the second derivative of a function in terms of the transform of its first derivative, then in terms of the transform of the function itself.

5.  Apply the method of variation of parameters to find the solution of a given second-order linear nonhomogeneous equation, given the solutions of the associated homogeneous equation.

 

1.   Analyze mechanical and electrical systems which can be modeled by homogeneous and nonhomeogeneous second-order linear differential equations with constant coefficients; model a given system, solve the equation, apply initial conditions and interpret the solution.

 

Module 3:  Systems of Linear Equations and Laplace Transforms

Assignments 24 - 34

Test #3 covers material through Assignment 34, and is due within a week after completing Assignment 34.

 

1.   Solve a linear system of two equations with constant coefficients for which only one independent solution results from the solution to the eigenvalue problem.

2.  Apply the periodicity property of the Laplace Transform to solve a nonhomogeneous second-order equation with constant coefficients which models a physical system driven by a square wave.

3.  Represent a linear system of differential equations as the product of a matrix, a column vector representing the solution and the derivative of that column vector.

4.  Apply calculus to matrix functions and products of matrix functions.

 

 

1.  For a given system of linear equations with initial conditions, find the largest interval over which a solution is guaranteed.

2.  For a given system of two or more first-degree linear equations find a higher-order equation whose solutions are equivalent.

3.  Apply the Wronskian to determine whether a given set of solutions of a linear system is a fundamental set.

 

 

1.  Apply Abel's Theorem to find the Wronskian of a given system for a given value of the dependent variable, given the value of the Wronskian at another given value of that variable.

2.  Verify whether a given set of solutions to a homogeneous system of linear differential equations is linearly independent.

3.  Show that the fundamental matrix (the psi matrix) for a given system can itself be interpreted as a solution of the system.

4.  Determine whether the product of the fundamental matrix for a given system, with another given matrix, results in a fundamental matrix.

5.  Find a fundamental matrix for a given system, and for a given initial condition on the fundamental matrix, use matrix operations to find the solution which satisfies the given condition.

 

 

1.  Verify whether a given set of solutions to a homogeneous system of linear differential equations is linearly independent.

2.  Show that the fundamental matrix (the psi matrix) for a given system can itself be interpreted as a solution of the system.

3.  Determine whether the product of the fundamental matrix for a given system, with another given matrix, results in a fundamental matrix.

4.  Find a fundamental matrix for a given system, and for a given initial condition on the fundamental matrix, use matrix operations to find the solution which satisfies the given condition.

 

 

1.  Solve linear systems of constant-coefficient differential equations having distinct real eigenvalues.

2.  Solve linear systems of constant-coefficient differential equations having complex conjugate eigenvalues.

3.  Solve mixing problems involving two or more tanks and constant flow rates.

4.  Solve vector differential equations.

 

 

1.  Solve linear systems of constant-coefficient differential equations having repeated eigenvalues.

2.  Use variation of parameters to solve nonhomogeneous linear systems.

 

 

1.  Given an n x n matrix determine whether it has n linearly independent eigenvectors.

2.  Given an n x n matrix with n linearly independent eigenvectors, use a change-of-basis matrix to perform the similarity transformation required to diagonalize it.

3.  Given a first-order linear system of differential equations with constant coefficients determine if the matrix associated with the system is diagonalizable.

3.  Given a first-order linear system of differential equations with constant coefficients, representable by a diagonalizable matrix, use the appropriate change-of-basis matrix to diagonalize the system and uncouple the equations, solve, and use the inverse of the change-of-basis matrix to transform the solution back into the original basis, and interpret to obtain the solution of the original system.

 

 

1.  Given a linear system of differential equations, find the fundamental matrix and use it to find the propagator matrix phi(t, s) = psi(t) psi^-1(s).

2.  Find the exponential, sine, cosine or square root of a diagonal matrix.

3.  Given a matrix and an associated full set of eigenpairs (eigenvalues and eigenvectors), ...

... matrices don't generally commute with their derivatives so we can't push the analogy too far ...

 

 

1.  Using the definition of the Laplace Transform find the transform and its domain for a given function.

 

 

1.  Use a table of Laplace Transforms to find the transforms of given functions, and/or to find inverse transforms of given expressions.

2.  Graph given combinations of Heaviside functions and determine their transforms.

3.  Combine multiples and shifts of Heaviside functions to obtain given periodic waveforms including but not limited to square and sawtooth patterns.

4.  Apply shift theorems

5.  Apply the method of partial fractions to obtain inverse Laplace Transforms.

6.  Solve initial given value problems using Laplace Transforms.

 

 

1.  Use Laplace Transforms to solve differential equations representing systems with periodic driving functions.

 

Play Windows Media

video_listing.htm

 

 

Specific objectives are also stated at the course homepage, where they are correlated assignment by assignment.

Requirement of communication

Regular communication is required of the student. This includes turning in assignments in a timely fashion and responding in a timely manner to feedback on these assignments. Any deviation of more than three days from the chosen schedule of the course must be approved in advance by the instructor. Exceptions will of course be made in the event of documented illness or other unexpected emergencies, but the instructor should be informed of such situations within a reasonable time of occurrence.

Getting Started in the Course

The Blackboard site for your course directs you to the Initial Information document and instructions for establishing communication, information on the nature of your course and the Orientation and Startup instructions you will need to understand and navigate the course.   This process will constitute appropriately the first week's assignments for your course (about the first half of the week during the shorter summer term), and will show you the basic navigation of the website including how to communicate, submit work, locate assignments and due dates, and more.

Text and Other Instructional Materials

The text is specified in Textbook Information, which the student will have encountered prior to arriving at this page.  Any student who has not noted Textbook Information is advised to review all information to be sure no other essential details have been missed.

Areas to be Covered

Units to be covered:

Chapters 6-11 inclusive, plus supplementary material posted by instructor.

Chapter Topics:

Chapters 1-3:  Introduction; First-order Equations (linear and nonlinear)

Chapter 4:  Second-order Linear Equations

Chapters 6 and 7 (Chapter 7 partial):  Systems of Linear Equations, Laplace Transforms

Specific information regarding assignments and areas covered is included on the homepage.

Instructional methods

Students will complete and submit the assignments specified on the homepage.

The instructor will respond in a timely fashion to any work submitted, making suggestions where improvement is needed and posing questions designed to enhance the student's learning experience. The student will be required to respond to all critiques, except those designated otherwise.

Questions posed by students and the instructor's responses will be posted to a site, specified in at the beginning of the course, for the student's review.

Students may on occasion be asked to critique work done by other students.  Full student anonymity will be preserved, with no reference  to the identity of any party in this exchange.

The instructor is available via web forms (to which you will be introduced at the very beginning of the course), and will normally respond by the end of the day following your submission (and more typically on the same day) with answers to properly posed questions, feedback on your efforts, and other information. Exceptions may occur in the event of Internet problems or other technical events. 

Use of email:  Prior to registration and receipt of initial instructions students my use Email to communicate with the instructor.  However email is much less reliable than web forms, and after registration and receipt of initial instructions anything sent through email should first be sent using the appropriate form.

Grading policy

Three tests will be administered.

A student's portfolio, consisting of instructor responses to assigned work and/or daily quizzes, will at the end of the term be assigned a grade.  A student who completes all assigned work in the prescribed manner can expect to make an A on this aspect of the course. The average of grades assigned on this work will count as 1/4 of a test grade. If this average is higher than the average on other tests, it will be counted as 1/2 of a test grade.

Raw test scores will be normalized to the following scale, according to the difficulty of the test, as specified in advance of each test by the instructor:

A: 90 - 100

B: 80 - 90

C: 70 - 80

D: 60 - 70

F: Less than 60.

The final grade will be a weighted average according to the above guidelines. A summary of the weighting is as follows:

Test #1:  Weight .5 or 1.0, to the advantage of the student

Test #2: Weight 1.0

Test #3: Weight 1.0

Assignment/Quiz Grade Average: Weight .25 or .5, to the advantage of the student.

The table below summarized the calculation of course grades:

assessment weighting contribution to total score
test 1 1 test score * 1
test 2 1 test score * 1
test 3 1 test score * 1
portfolio 1/4 <= p_weight <= 1/2 portfolio score * p_weight
  total of weightings total of contributions

Final average = total of contributions / total of weightings

Criteria for Grading of Tests:

Tests will consist of problems designed to measure the level of your achievement of the course goals. 

Each problem is graded on a 10-point scale, with the following guidelines:

In the event of a college-wide emergency 

In the event of a College-wide emergency, course requirements, classes, deadlines, and grading schemes are subject to changes that may include alternative delivery methods, alternative methods of interaction with the instructor, class materials, and/or classmates, a revised attendance policy, and a revised semester calendar and/or grading scheme. 

In the case of a College-wide emergency, please refer to the following about changes in this course:

 

Instructor’s email dsmith@vhcc.edu (however, you should use your access page for the most reliable responses) 

For more general information about the emergency situation, please refer to:

 

·      Web site  - www.vhcc.edu

·      Telephone Number - 276-739-2400

·         Emergency Text Messaging or Phone System- Virginia Highlands Community College uses VHCC Alert to immediately contact you during a major crisis or emergency. VHCC Alert delivers important emergency alerts, notifications and updates to you on your E-mail account (work, home, other), cell phone, pager or smartphone/PDA (BlackBerry, Treo & other handhelds). VHCC Alert is a free service offered by VHCC. Your wireless carrier may charge you a fee to receive messages on your wireless device. VHCC will test the alert system each semester. Register online at alert.vhcc.edu or by sending a text message to 411911 keyword: VHCC 

In the event of an emergency just regarding this class, the instructor will contact all students via email, and may post information to your access site.  You should check both email and your access site.