# Genesee Course Listing

## Official Course Information

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### Mathematics Courses:

#### MAT255 - Differential Equations

Credits: 4

Catalog Description: Covers solutions of first order differential equations, Euler's Method, linear differential equations with constant coefficients, non-linear equations, LaPlace transforms, numerical solutions, and series solutions. Applications focus on modeling with first- and second-order linear equations. CAS Software (such as Derive or Maple) will be used for solving a variety of application problems. Spring only. Prerequisite: MAT 142 with a grade of āCā or higher or by placement.

Lecture: 4 hrs.

Course Learning Outcomes (CLOs):

A student who successfully completes Differential Equations will be able to do each of the following in an exam setting and/or through the use of a Computer Algebra System:

1. Show that a function of the form y = y(x) is a valid solution to a given differential equation.

2. Find a solution to any first order differential equation (DE) that is

a.Separable

b.Linear

c.Exact

d.Homogeneous

e.Bernoulli*

3. Use Numerical Techniques (i.e. Euler's Method) to Approximate Solutions to First Order DE's

4. Solve application problems related to first order DE's.

5. Find a solution to a second order linear DE using

a.Reduction of Order formula

b.Method of Constant Coefficients

c.Method of Annihilators

d.Variation of Parameters

e.Cauchy-Euler Technique

6.Solve application problems related to second order DE's.

7.Find a Power Series solution to a DE with polynomial coefficients

8.Use a Laplace transform (and inverse transform) to solve a DE

9.Solve a system of Linear DE's by Elimination

* This course objective has been identified as a student learning outcome that must be formally assessed as part of the Comprehensive Assessment Plan of the college. All faculty teaching this course must collect the required data and submit the required analysis and documentation at the conclusion of the semester to the Office of Institutional Research and Assessment.

Content Outline:

1. Introduction to Terminology and Modeling with Differential Equations

2. First Order Differential Equations

a) Direction Fields

b) Separable equations

c) Linear equations

d) Exact equations

e) Substitution Methods

f) Numerical Techniques (Euler's Method)

3. Applications of First Order Differential Equations

a) Applications of first order linear equations

b) Applications of first order non-linear equations

c) Applications of systems of linear equations

4. Second Order Linear Equations

a) Fundamental solutions of the homogeneous equation

b) Reduction of order

c) Homogeneous equations with constant coefficients

d) Non-homogeneous equations

i) Method of undetermined coefficients

ii) Method of variation of parameters

e) Cauchy-Euler equation

f) Solving Systems of Linear Equations by Elimination

g) Nonlinear Differential Equations

5. Applications of Second Order Differential Equations

6. Differential Equations with Variable Coefficients

a) Power series solutions around ordinary points

b) Series solution around singular points

7. Laplace Transform

a) Definition of Laplace transform & inverse transforms

b) Solution of initial value problems

c) Transforms of derivatives and integrals

d) Transforms of periodic functions

8. Systems of Linear First-Order Differential Equations

a) Homogeneous Linear Systems

b) Non-Homogeneous Linear Systems

Effective Term: Fall 2012