ModelCourses/Ordinary Differential Equations
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Contents
General Description
- Sophomore or junior level course
- Pre-requisite: Calculus
Possible textbooks include, but are not limited to:
- Fundamentals of Differential Equations and Boundary Value Problems, by Nagle, Saff, and Snider, Addison-Wesley.
- Elementary Differential Equations and Boundary Value Problems by Boyce and DiPrima, Wiley.
Course Objectives
Upon successful completion of this course, students will be able to:
- Determine if a given function is a solution to a particular differential equation; apply the theorems for existence and uniqueness of solutions to differential equations appropriately;
- Distinguish between
(a) linear and non-linear differential equations;
(b) ordinary and partial differential equations;
(c) homogeneous and non-homogeneous differential equations; - Solve ordinary differential equations and systems of differential equations using:
(a) Direct integration
(b) Separation of variables
(c) Reduction of order
(d) Methods of undetermined coefficients and variation of parameters
(e) Laplace transform methods - Determine particular solutions to differential equations with given initial conditions.
- Analyze real-world problems such as motion of a falling body, compartmental analysis, free and forced vibrations, etc.; use analytic technique to develop a mathematical model, solve the mathematical model and interpret the mathematical results back into the context of the original problem.
- Apply matrix techniques to solve systems of linear ordinary differential equations with constant coefficients.
- Find the general solution for a first order, linear, constant coefficient, homogeneous system of differential equations; sketch and interpret phase plane diagrams for systems of differential equations.
Problem Sets
Use of Problem Sets
The problem sets were assembled to allow for personalization by individual faculty. The topics covered are fairly standard in an introductory Differential Equations course, but faculty can rearrange the topics and delete any sections they do not wish to cover, or wish to assess by other means. The names of the problem sets are meant to be descriptive and the learning objectives will help you evaluate if the set should be included or not.
Download the problem sets
A copy of the course can be found at at the MAA website
The course can be downloaded here <insert link>.
To use the files remove the .txt from the end. The .tgz can be added. This file can now be directly uploaded into your own course:
- go to Filemanager
- Upload the file
- etc <provide enough detail to allow for easy installation by anyone>
Description of Problem Sets
- Chapter 0 : Introduction
- 01 : Solutions and Initial Value Problems
Students will be able to:- Determine if a given function is a solution to a particular differential equation
- Identify elements of a given solution to a differential equation
- Model with differential equations
- 02 : Classification of Differential Equations
Students will be able to distinguish between:- linear and non-linear differential equations;
- ordinary and partial differential equations;
- homogeneous and non-homogeneous higher order differential equations
- 01 : Solutions and Initial Value Problems
- Chapter 1 : First-Order Equations
- 01 : Integrals as Solutions
Students will be able to:- Solve simple first order differential equations by direct integration
- Solve simple differential equations using their knowledge from Calculus
- 02 : Existence and Uniqueness Theorem
Students will be able to:- Apply the theorem for existence and uniqueness of solutions to differential equations appropriately;
- If necessary, re-write the first order equation in a form so the Existence and Uniqueness Theorem can be applied;
- Determine the largest domain where a given first order differential equation has unique solution;
- Check the sufficient conditions for a first order linear differential equation to have a unique solution about the initial value.
- 03 : Slope Fields and Equilibrium Solutions
Students will be able to:- Empty
- 04 : Separable Equations
Students will be able to:- Identify separable equations;
- Use separation of variables to find the general solution of separable equations;
- Find the solution of initial value problems;
- Find the constant solutions, if any, and appropriately use the initial condition;
- Solve applications modeled by separable equations.
- 05 : Linear Integrating Factors
Students will be able to:- Identify first order linear equations;
- Write a first order linear equation in standard form to be used with integrating factor;
- Use the method of integrating factor to solve first order linear equations;
- Solve applications of first order linear equations;
- Identify the largest domain where the initial value problem first order linear equation has unique solution;
- Solve applications modeled by first order linear equations.
- 06 : Exact Equations
Students will be able to:- Identify exact equation by using the exactness test;
- Apply the method for solving exact equations.
- 07 : Substitution Techniques and Integrating Factors
Students will be able to:- Use substitution techniques to solve certain types of equations such as Bernoulli and first order homogeneous;
- Apply the method of integrating factor to transform equations into forms that are easily solvable, such as exact equations;
- Identify and solve Bernoulli equation;
- Identify and solve first order homogeneous equations.
- 08 : Autonomous
Students will be able to:- Match autonomous differential equations with a given phase line graphs;
- Find bifurcation value(s) for given one-parameter family of autonomous equations.
- 09 : Euler's Method
Students will be able to:- Estimate solutions to differential equations using Euler’s method;
- Identify the impact of step size of the accuracy of estimated solutions;
- Identify if the estimated solution is expected to be an overestimate or underestimate;
- Estimate solutions using Improved Euler’s method.
- 10 : Applications
Students will be able to:- Model with and solve first order equations such as the ones based on Newton's Law of Cooling and Heating, Compartmental Analysis, Exponential Growth and Decay.
- 11 : Miscellaneous
Students will be able to:- answer/solve various questions that may be either challenging problems, extending standard ideas or questions not usually a part of a standard curriculum.
- Empty
- 01 : Integrals as Solutions
- Chapter 2 : Higher-Order Equations
- 01 : Introduction to Linear
Students will be able to: - 02 : Linear 2nd-Order Constant-Coefficient Homogeneous Equations
Students will be able to:- Determine if a given set of functions is a fundamental set of solutions of a homogeneous second order linear equation with constant coefficients;
- Solve the auxiliary equation to create the fundamental set of solutions;
- Recognize the type of general solution based on the roots of the auxiliary equation.
- 03 : Linear Higher-Order Constant-Coefficient Homogeneous Equations
Students will be able to:- Determine if a given set of functions is a fundamental set of solutions of a homogeneous second or higher order linear equations with constant coefficients;
- Solve the auxiliary equation to create the fundamental set of solutions;
- Recognize the type of general solution based on the roots of the auxiliary equation (real, complex, repeated).
- 04 : Free Mechanical Vibrations
Students will be able to: - 05 : Nonhomogeneous Undetermined Coefficients
Students will be able to: - 06 : Nonhomogeneous Variation of Parameters
Students will be able to:- Empty
- 07 : Forcing Resonance
Students will be able to:- Empty
- 01 : Introduction to Linear
- Chapter 3 : Linear Systems
- 01 : Intro to Systems
Students will be able to:- understand basic language and concepts of linear algebra as applied to systems of differential equations.
- 02 : Matrices
Students will be able to:- convert a system of linear differential equations to its matrix representation
- convert the matrix representation of a system of differential equations to a system
- 03 : Linear Systems
Students will be able to: - 04 : Eigenvalue Method
Students will be able to:- solve a linear system using eigenvalues and eigenvectors
- understand a geometric interpretation of eigenvectors
- 05 : 2D Systems Vector Fields
Students will be able to: - 06 : Second Order Systems Mass Spring Systems
Students will be able to: - 07 : Repeated Eigenvalues
Students will be able to:- solve a linear system using eigenvalues and eigenvectors when the eigenvalues are repeated
- understand a geometric interpretation of eigenvectors when the eigenvalues are repeated
- 08 : Complex Eigenvalues
Students will be able to:- solve a linear system using eigenvalues and eigenvectors when the eigenvalues are complex
- understand a geometric interpretation of eigenvectors when the eigenvalues are complex
- 09 : Nonhomogeneous Systems
Students will be able to:
- 01 : Intro to Systems
- Chapter 4 : Laplace Transforms
- 01 : Laplace-transforms
Students will be able to: - 02 : Shifts-and-IVPs
Students will be able to: - 03 : Partial-fractions
Students will be able to: - 04 : Periodic-functions
Students will be able to: - 06 : Convolution
Students will be able to: - 07 :Delta-function
Students will be able to:
- 01 : Laplace-transforms
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