# MATH-288: Differential Equations

School
Science, Technology, Engineering and Math
Division
Mathematics
Department
Mathematics
Course Subject
Mathematics
Course Number
288
Course Title
Differential Equations
Cross-Referenced Course
Credit Hours
5.00
Instructor Contact Hours Per Semester
77.00 (for 15-week classes)
Student Contact Hours Per Semester
77.00 (for 15-week classes)
A-E
Pre-requisites
MATH-280 with a C or better OR concurrent enrollment in MATH-280
Catalog Course Description

Introduces ordinary differential equations by means of algebraic, numerical, and graphical analysis (including phase-plane analysis). Examines first order differential equations, second and higher order linear equations, methods for nonhomogeneous second order equations, series solutions, Laplace transforms, linear systems, and linearization of nonlinear systems. Covers various applications throughout the course. Requires a non-graphing scientific calculator for some formal assessments, access to a free graphing utility application to support learning and during informal assessments, and access to an online homework assessment.

## Goals, Topics, and Objectives

Goal Statement
1. To provide an introduction to the nature and significance of differential equations for students of engineering, mathematics, and science.
2. To demonstrate various applications of differential equations to problems from the physical sciences and engineering.
3. To provide methods for solving differential equations that have proved useful in a wide variety of applications.
4. To present an exposition of differential equations that incorporates algebraic, numerical and graphical analysis, without undue emphasis on theoretical abstraction or routine mechanical manipulation.
5. To use technology to graph solutions of ordinary differential equations (ODEs) and to do explorations and projects involving ODEs.
Core Course Topics
1. Introduction to Differential Equations
1. Determine the order of a differential equation and whether it is linear or nonlinear.
2. Verify that a function or family of functions solves a differential equation.
3. Determine parameter values such that a member of a family of solutions solves a given initial-value problem.
2. First-Order Differential Equations
1. Relate a solution curve to the direction field for a differential equation.
2. Solve a separable differential equation.
3. Solve a first-order linear differential equation.
4. Solve an exact differential equation.
5. Use an appropriate substitution to rewrite and solve a differential equation.
6. Apply Euler’s method to obtain a numerical approximation of a differential-equation solution-function value.
3. Higher-Order Differential Equations
1. Verify that a set of functions is a fundamental set of solutions of a differential equation.
2. Use reduction of order to find a second solution of a second-order differential equation given one solution.
3. Solve a homogeneous linear differential equation with constant coefficients.
4. Apply the method of undetermined coefficients to solve a nonhomogeneous linear differential equation with constant coefficients.
5. Apply the method of variation of parameters to solve a linear second-order differential equation.
6. Solve a Cauchy-Euler equation.
4. Series Solutions of Linear Equations
1. Find a Taylor-series solution of an initial-value problem.
2. Find power-series solutions of a differential equation about an ordinary point.
5. Modeling with Differential Equations
1. Solve a problem in the physical sciences (such as a growth or decay problem, a mixture problem, or a Newton’s Law of Cooling problem) whose solution utilizes a first-order linear differential equation.
2. Solve a problem in the physical sciences (such as a spring/mass-system problem) whose solution utilizes a second-order linear differential equation.
3. Solve a problem in the sciences (such as a logistic-growth problem) whose solution utilizes a nonlinear differential equation.
4. Solve a problem in the sciences (such as a connected-mixing-tanks problem or a predator-prey problem) whose solution utilizes a system of differential equations.
6. The Laplace Transform
1. Calculate the Laplace transform of a function using the definition of Laplace transform.
2. Use algebraic manipulation or a table to find the Laplace transform of a function or its derivatives.
3. Use algebraic manipulation or a table to find an inverse Laplace transform.
4. Use the Laplace transform to solve an initial-value problem.
5. Use the Laplace transform to solve a system of linear differential equations.
7. Systems of Differential Equations
1. Solve a system of linear differential equations by elimination.
2. Solve a system of linear differential equations by elimination using the Laplace transform.
3. Solve a system of linear differential equations using the eigenvalue method.
4. Analyze the phase plane for a linear system.
5. Linearize a nonlinear system of differential equations.

## Assessment and Requirements

• All students will be required to complete a comprehensive final examination that assesses the learning of all course objectives. This final exam must be weighted in a manner so that this exam score is worth a minimum of fifteen percent (15%) of the final course grade. In selected semesters this final exam may be a common final exam administered to all sections of Math-288.
• All students will be required to complete at least two proctored on-campus exams. The cumulative value of those exams must be at least 40% of a student’s final grade.
• All students will be required to complete online homework. This online homework must be weighted in such a manner so that it is worth between six percent (6%) and twelve percent (12%) of the final course grade.
• Additional assessment of student achievement may include assignments, quizzes, and exams.
• For proctored in-person formal assessments (quizzes, tests, and exams) the only technology students can use is a non-graphing scientific calculator. Quizzes, tests, and exams may have non-calculator parts. Class projects and informal assessments will require students to use a free graphing application to support learning.
• Application problems must not only be included on chapter exams but also on the final exam.
General Course Requirements and Recommendations
• A non-graphing scientific calculator is required for proctored in-person formal assessments.
• Free graphing applications will be used to support learning, informal assessments, and class work.
• Application problems must be covered in all mathematics courses. Every section in any course outline that includes application problems must be covered.

## Outcomes

General Education Categories
• Mathematics
Institutional Outcomes
• Quantitative Literacy
MTA Categories
• Category 3: Mathematics
• Category 3: Mathematics (College Algebra Track)
Satisfies Wellness Requirement
No
Effective Term
Fall 2024