# ModelCourses/Ordinary Differential Equations

## 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.
- A modern introduction to differential equations, 2nd Ed., Henry J. Ricardo, Elsevier, 2009.
- Differential Equations, 4th Ed., Blanchard, Devaney, Hall, 2012.

## 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

### 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 : 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.

**03 : 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.

**04 : Slope Fields and Equilibrium Solutions**

Students will be able to:- Match given direction fields (slope fields, vector fields) with given first-order differential equations

**05 : Exact Equations**

Students will be able to:- Identify exact equation by using the exactness test;
- Apply the method for solving exact equations.

**06 : 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.

**07 : 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.

**08 : 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.

**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.

### Chapter 2 : Higher-Order Equations

**01 : Introduction to Linear**

Students will be able to:- Recognize higher-order linear equations.
- Use the principle of superposition to write the general solution of a linear equation, given a basis of solutions.
- Verify is a set of given solutions is linearly independent.
- Compute the Wronskian associated with a set of solutions.
- Derive the characteristic equation corresponding to a linear equation with constant coefficients.

**02 : Linear 2nd-Order Constant-Coefficient Homogeneous Equations**

Students will be able to:- 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 higher order linear equation;
- 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:- Create a differential equation model corresponding to a physical system;
- Find the general solution of the equation and solve initial value problems;
- Relate the solution to the physical system;
- Classify systems as underdamped, overdamped or critically damped.

**05 : Nonhomogeneous Undetermined Coefficients**

Students will be able to:- Use the principle of superposition to find solutions of the nonhomogeneous equation;
- Characterize the general solution of a nonhomogeneous equation;
- Find a particular solution using the method of undertermined coefficients;
- Solve the corresponding initial value problem.

**06 : Nonhomogeneous Variation of Parameters**

Students will be able to:- Use variation of parameters to find a particular solution of the nonhomogeneous equation.
- Solve the corresponding initial value problem.

**07 : Forced Linear Oscillators and Resonance**

Students will be able to:- Create a differential equation corresponding to a forced linear oscillator.
- Find the general solution of a linear oscillator with sinusoidal forcing.
- Write the general solution in phase-amplitude form.
- Characterize the phenomenon of resonance and identify the resonant frequency of a system.

### Chapter 3 : Linear Systems

**01 : Intro to Systems**

Students will be able to:- Translate a system of differential equations into matrix representation;
- Solve a simple system of differential equations by direct substitution;
- Set up a word problem involving systems of differential equations.

**02 : Matrices**

Students will be able to:- Find intervals of invertibility for matrix functions;
- Find derivatives and antiderivatives of matrix functions;
- Calculate the inverse of a matrix function.

**03 : Linear Systems**

Students will be able to:- Translate systems of differential equations into matrices;
- Use substitution to convert higher order differential equations into linear systems;
- Verify solutions to systems of differential equations;
- Use the Wronskian to determine linear independence of solutions.

**04 : Eigenvalue Method**

Students will be able to:- Compute eigenvalues and eigenvectors for a given matrix;
- Find the general solution for linear systems of homogeneous differential equations;
- Find the particular solution for linear systems of homogeneous differential equations with initial values.

**05 : 2D Systems Vector Fields**

Students will be able to:- Identify the vector field corresponding to a homogeneous system of linear differential equations;
- Identify sinks and sources;
- Sketch a particular solution for a homogeneous system of linear differential equations.

**06 : Second Order Systems Mass Spring Systems**

Students will be able to:- Solve two-body spring and mass systems;
- Solve two-tank mixing solution problems.

**07 : Repeated Eigenvalues**

Students will be able to:- Find general solutions for systems with repeated eigenvalues;
- Find particular solutions for systems with repeated eigenvalues;
- Illustrate an understanding of the relationship between coefficient matrices and phase planes.

**08 : Complex Eigenvalues**

Students will be able to:- Find general solutions for systems with complex eigenvalues;
- Find particular solutions for systems with complex eigenvalues;
- Illustrate an understanding of the relationship between coefficient matrices and phase planes.

**09 : Nonhomogeneous Systems**

Students will be able to:- Understand and apply the superposition principle.
- Analyse the stability of a nonhomogeneous system and its equilbria.
- Solve systems nonhomogeneous of linear differential equations;
- Find equilibrium solutions for systems of nonhomogeneous linear differential equations;
- Apply methods of solution to two-tank mixture problems.

### Chapter 4 : Laplace Transforms

**01 : Laplace-transforms**

Students will be able to:- Determine whether or not a function has a Laplace transformation;
- Find the Laplace transform of a function by integration;
- Find the Laplace transform of a function using a table;
- Identify the domain for the Laplace transform of a given function;
- Find the inverse Laplace transform of a function using a table.

**02 : Shifts-and-IVPs**

Students will be able to:- Express piecewise functions using the Heaviside function;
- Find the Laplace transform of a function which includes the Heaviside function;
- Find the inverse Laplace transform of a shifted function;
- Apply Laplace transformations to the solution of non-homogenous IVPs;
- Apply Laplace transformations to the solution of an IVP with a piecewise non-homogeneous component. (may be moved to a new section dedicated to solving IVPs)

**03 : Partial-fractions**

Students will be able to:- Apply partial fraction decomposition to a rational function;
- Find the inverse Laplace transformation of a rational function using partial fraction decomposition.
- Apply Laplace transformation and partial fraction decomposition techniques in order to solve IVPs.

**04 : Periodic-functions**

Students will be able to:- Find the Laplace transform of a periodic piecewise function;
- Construct and solve an IVP from a word-problem requiring a periodic piecewise function.

**06 : Convolution**

Students will be able to:- Find the convolution of two functions by direct integration;
- Find the convolution of two functions using the Laplace transform and its inverse;
- Solve IVPs that include convolutions.
- Solve integral and integro-differential equations.

**07 :Delta-function**

Students will be able to:- Evaluate integrals involving the Delta function;
- Solve IVPs that include the Delta function.

WeBWorK Workshop, Ann Arbor Michigan June 2013, PREP Workshop Washington D.C. July 2013