MTH 289 Differential Equations Extended
Course Title: MTH 289 Differential Equations Extended
Course Description
Presents systems of differential equations, power series solutions, Fourier series, Laplace transform and Fourier transform, partial differential equations, and boundary value problems. Designed as a math elective course for mathematical, physical, and engineering science programs. Lecture 3 hours per week. 3 credits.
General Course Purpose
The purpose of the course is to provide for a smooth transition of STEM students to 4year colleges and introduce them to advanced topics of mathematics, physics and engineering: numerical methods for solving differential equations, classical partial differential equations, methods for solving PDEs and boundaryvalue problems (BVPs).
Course Prerequisites/Corequisites
Prerequisite: Completion of MTH 267 Differential Equations with a grade of C or better or equivalent.
Course Objectives
Upon completing the course, the student will be able to:
System of Linear First Order Differential Equations

Define system of linear firstorder differential equations, Initial value problem (IVP) and its solution vector, linear dependence/independence, fundamental set of solutions

Check that a vector of functions is a solution of a system or an initial value problem (IVP)

Apply criterion for linearly independent solutions and find general solution for homogeneous and nonhomogeneous systems (for the 3 types of eigenvalues: distinct real, complex, repeated)

Solve nonhomogeneous linear systems by the methods of undetermined coefficients and variation of parameters
Numerical solutions of Ordinary Differential Equations

Understand the concept of local and global truncation errors, stability of numerical method

Use singlestep and multistep methods( Euler’s Method, Improved Euler’s Method, 1st, 2nd and 4thorder RungeKutta Method, AdamsBashforthMoulton Method) and finite difference method to approximate ivp and bvp solutions
Plane Autonomous Systems

Explain the concept of autonomous systems of firstorder des (linear and nonlinear)

Find critical points and classify critical points of linear and nonlinear systems (stable/unstable nodes, saddle point, degenerate unstable node, center, stable/unstable spiral points), identify equilibrium solution and periodic solution

Use stability criterion for plane autonomous systems

Apply the concept of linearization of differentiable function and classify stable and unstable critical points

Perform stability analysis for linear/nonlinear systems for various applications
Orthogonal Functions

Define orthogonal functions and sets of orthogonal functions

Write the definition of the Fourier Series and expansion of functions in a Fourier Series

Define SturmLiouville problem and solve it

Write the definitions and expand the function in FourierBessel Series
BoundaryValue Problems in Rectangular Coordinates

Define linear/nonlinear, homogeneous/nonhomogeneous partial differential equations

Classify the linear secondorder pdes as hyperbolic, parabolic or elliptic

Use the method of separation of variables to find particular solution of pdes

Identify classical and modified pdes and bvps (1d heat equation, 1d wave equation and 2d form of Laplace’s Equation) and solve them

Use the concept of orthogonal series expansions or generalized Fourier Series and solve bvps by using orthogonal series expansions
Integral Transforms

Find the Laplace transform of partial derivatives of functions of two variables, use Laplace transform to solve bvps

Define a Fourier integral of function and conditions for convergence, the Fourier integral of even/odd functions

Use the definitions of three Fourier transform pairs (direct and inverse integral transforms)

Solve bvps using the Fourier transform
Major Topics to be Included
System of Linear First Order Differential Equations
Numerical solutions of Ordinary Differential Equations
Plane Autonomous Systems
Orthogonal Functions
BoundaryValue Problems in Rectangular Coordinates
Integral Transforms