Department Mathematics
Subject Area and Course Number
MATH 220
Title Differential Equations
Disciplines
Mathematics (Masters Required)
Units
4.000

Repeatability 0 - May not be repeated
Catalog Course Description
An introductory course on the theory and applications of ordinary and partial differential equations with studies of constant coefficient equations, various series techniques, introduction to Laplace Transforms, qualitative and quantitative solutions to linear and nonlinear systems of differential equations, and separable partial differential equations. This course is intended for STEM majors who have successfully completed both Math 200: Multivariable Calculus and Math 210: Linear Algebra. Students in Math 220 will demonstrate their learning of course outcomes through multiple methods of assessment, including at least 4 hours of controlled assessments in the form of in-person proctored exams/project presentations, and a cumulative in-person proctored final exam.
Lecture Hours
64.000-72.000 Total Hours
Lab Hours
Total Hours
Out-of-Class-Hours
128.000-144.000 Total Hours
Total Contact Hours
64.000-72.000 Total Hours

Prerequisite: MATH 200 and MATH 210
Prerequisite or Corequisite: None
Concurrent Corequisite: None
Course Advisories: None
Limitation on Enrollment: None

Course Objectives:
Identify the type of a given differential equation and select the appropriate analytical technique for finding the solution of first order and selected higher order ODE's and PDE's.
Write differential equations to represent some types of natural phenomena.
Use transform techniques in problems involving discontinuous and impulsive forcing functions.
Apply matrix techniques to solve systems of linear differential equations.
Solve differential equations using series techniques.
Analyze critical points and stability for systems of equations.
Apply the Existence - Uniqueness theorem for ODE's.

Student Learning Outcomes
MATH220 SLO1 - Apply Differential Equations to problems in the sciences and solve various linear and nonlinear ODE’s analytically or numerically.
MATH220 SLO2 - Use Laplace transforms to solve second order linear ODE’s with discontinuous forcing functions or impulse functions.
MATH220 SLO3 - Determine the qualitative behavior of an autonomous nonlinear system by means of an analysis of behavior near critical points.
MATH220 SLO4 - Compute Fourier coefficients, and find periodic solutions of linear ODE's and PDE's by means of Fourier series and separation of variables.

Course Content and Scope
  1. First Order Differential Equations
    1. Existence and Uniqueness Theorems
    2. Solution of linear equations
    3. Separable Equations
    4. Exact Equations
    5. Substitute to solve bernoulli and homogeneous equations
    6. Applications such as circuits, mixture problems, population modeling, orthogonal trajectories, and slope fields.
  2. Second and higher Order Linear Equations
    1. Homogeneous equations  
    2. Linear independence of solutions;Wronskian; Fundamental Solutions
    3. Nonhomogeneous equations; undertermined coefficients and variation of parameters
    4. Applications such as circuits and harmonic oscillators
  3. Series Solutions–Variable coefficients
    1. Ordinary points
    2. Regular singular points
    3. Bessel functions
  4. Laplace transforms
    1. Initial value problems
    2. Step functions
    3. Impulse functions
    4. Convolutions
  5. Systems of first order linear equations
    1. Eigenvalue–Eigenvector method for solving homogeneous systems with constant coefficients
    2. Fundamental matrices and matrix exponential
    3. Nonhomogeneous systems
  6. Partial Differential Equations
    1. Fourier Series
    2. Separable equations
    3. The heat equation
    4. The wave equation
    5. Laplace's equation
  7. Numerical Methods
    1. Euler's Method
    2. Runge-Kutta Method
  8. Nonlinear Differential Equations and Systems
    1. Stability and the Phase Plane
    2. Linear and almost linear systems

Methods of Instruction
Directed Study
Discussion
Lab
Lecture
Projects
Other Methods
Central instructional techniques of the course include live or recorded lectures with guided discussion and practice, problem-solving, contextualized examples, and readings of relevant text material. Print or digital lecture and discussion materials will be provided. To support student learning, students may be asked to apply concepts from instruction to written problem sets, projects, structured group activities, CAS problem sets, exam review materials, as well as to spend time reviewing and incorporating their individual student feedback from the instructor. These activities may take place either in or outside of class sessions, or both. Clearly defined expectations for all learning activities, due dates for assignments, and dates of exams will be available with advanced notice through the college's learning management system, the section syllabus, and in-class announcements.

Examples of assignments and/or activities (required reading and writing):
Solve the one dimensional heat equation with Dirichlet boundary conditions.
Representative of the types of assignments/outside-of-class assignments:

1. Readings/Viewings: Students will be directed to read chapters of the textbook, review digital or printed lecture materials, or view closed-captioned video recordings.

2. Discussion: Students will engage in class discussions with the instructor and their peers. Students are encouraged to question, respond, and work collaboratively toward achieving the learning goals of class sessions.

3. Written Assignments and Reflection: Students will spend a sufficient amount of time to work on mathematical problem sets requiring the understanding of abstract ideas and to practice techniques learned in lectures, as well as read or view and review course materials, assignments, assessments, and feedback to reflect upon their learning.  

4. Assignments that Demonstrate Critical Thinking: Students will demonstrate critical thinking skills in a variety of ways, such as analyzing mathematical information and visuals, critically examining logical arguments, recognizing concepts in various contexts, applying abstract knowledge to their own fields of study, solving problems in various disciplines, and/or utilizing technology (computer software/graphing calculator) within the context of the course concepts.

5. In-person Proctored Exams: Students will participate in multiple in-person proctored exams, including a cumulative final exam. Students will demonstrate their individual mastery of the course’s core concepts, dexterity with the mathematical techniques, and critical thinking and mathematical writing skills developed over the course of the semester.   

Method Of Evaluation
A student's grade will be based on multiple methods of assessment that incorporate mathematical modeling, preparation and analysis of visuals and graphs, dexterity with arithmetic skills, course specific computational skills, familiarity with mathematical vocabulary, problem-solving, critical thinking, and analysis of logical arguments. Included in such assessments are in-person proctored exams/projects, and a 2 hour, in-person proctored, cumulative final exam, each linked to the learning outcomes and objectives of the course and, combined, comprise at least 50% of the course grade. In addition, instructors may make use of frequent low-stakes assessments such as quizzing, written homework assignments, CAS/technology assignments, study guides and reviews, collaborative group work, discussion, and self-reflection.

Appropriate Texts and Supplies:
  • Differential Equations with Boundary-Value Problems Zill , Cengage Publishing, 2023.
  • Notes on Diffy Qs; Differential Equations for Engineers Lebl, Jiri, Creative Commons, 2016.
Course Fees and Other Materials:
  • TI-84 Graphing Calculator; digital or printed lecture materials are provided; access to Maple or an equivalent computer algebra system is provided.

Created On 05/23/2024
Board of Trustees: 05/22/2025
CAC Approval: 04/02/2019
CAC Approval: 04/21/2025
CAC Approval: 04/18/2019
CAC Approval: 04/21/2019