Theory and applications of ordinary and partial differential equations. Topics include constant coefficient equations, series techniques, introduction to Laplace Transforms, qualitative and quantitative solutions to linear and nonlinear systems of differential equations, and separable partial differential equations.
64.000-72.000 Total Hours
Total Hours
128.000-144.000 Total 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.
MATH220 SLO2 - Solve various linear and nonlinear ODE’s analytically or numerically.
MATH220 SLO3 - Determine the qualitative behavior of an autonomous nonlinear system by means of an analysis of behavior near critical points.
MATH220 SLO4 - Use Laplace transforms to solve second order linear ODE’s with discontinuous forcing functions or impulse functions.
MATH220 SLO5 - Compute Fourier coefficients, and find periodic solutions of linear ODE's and PDE's by means of Fourier series and separation of variables.
First Order Differential Equations
Existence and Uniqueness Theorems
Solution of linear equations
Separable Equations
Exact Equations
Substitute to solve bernoulli and homogeneous equations
Applications such as circuits, mixture problems, population modeling, orthogonal trajectories, and slope fields.
Second and higher Order Linear Equations
Homogeneous equations
Linear independence of solutions;Wronskian; Fundamental Solutions
Nonhomogeneous equations; undertermined coefficients and variation of parameters
Applications such as circuits and harmonic oscillators
Series Solutions–Variable coefficients
Ordinary points
Regular singular points
Bessel functions
Laplace transforms
Initial value problems
Step functions
Impulse functions
Convolutions
Systems of first order linear equations
Eigenvalue–Eigenvector method for solving homogeneous systems with constant coefficients
Fundamental matrices and matrix exponential
Nonhomogeneous systems
Partial Differential Equations
Fourier Series
Separable equations
The heat equation
The wave equation
Laplace's equation
Numerical Methods
Euler's Method
Runge-Kutta Method
Nonlinear Differential Equations and Systems
Stability and the Phase Plane
Linear and almost linear systems
Methods of Instruction
Directed Study
Discussion
Lab
Lecture
Projects
A combination of lecture and computer explorations will be used in the course. Students will be exposed to software and graphical approaches to the subject as well as to material from texts.
Solve the one dimensional heat equation with Dirichlet boundary conditions.
Appropriate Readings: Students are required to read assigned chapters in texts.
2. Writing Assignments: Students must work assigned mathematical problems requiring the understanding of abstract ideas.
3. Appropriate Outside Assignments: Students will be expected to spend a sufficient amount of time outside of class to practice techniques taught during class time, read assigned materials, and complete frequent homework and computer assignments.
4. Appropriate Assignments that Demonstrate Critical Thinking: Students must demonstrate mathematical skills which involve analyzing information, recognizing concepts in new contexts, and drawing analogies. They must also analyze logical arguments for validity and write proofs of their own using both inductive and deductive reasoning within a logical system. Students will also learn to use software tools in solving differential equations.
A student's grade will be based on multiple measures of performance in the solving of problems, designing of mathematical models, preparation and analysis of graphs, and analysis of logical arguments. Such measures will include at least four one-hour exams and a comprehensive final examination requiring demonstrations of problem solving skills. In addition, instructors may make use of quizzes, written homework assignments, computer assignments, or other appropriate means to judge a student's dexterity with mathematical skills and familiarity with mathematical vocabulary and methods of proof.
Differential Equations with Boundary-Value ProblemsZill , Cengage Publishing, 2017Notes on Diffy Qs; Differential Equations for EngineersLebl, Jiri, Creative Commons, 2016
Maple, Mathematica, Matlab, or equivalent computer algebra system