Math 246 Description and Prerequisites

UM Undergraduate Catalog Course Description

Math 246, Differential Equations for Scientists and Engineers (3 credits) This course is an introduction to ordinary differential equations. The course introduces the basic techniques for solving and/or analyzing first and second order differential equations, both linear and nonlinear, and systems of differential equations. Emphasis is placed on qualitative and numerical methods, as well as on formula solutions. The use of a mathematical software system (MSS), either Mathematica or MATLAB, is an integral and substantial part of the course. More detailed outlines of some of the topics to covered can be found below and at .

Course Prerequisites

A More Detailed Outline

Introduction to and Classification of Differential Equations
First Order Equations
     Linear, separable and exact equations
     Introduction to symbolic solutions using a MSS
     Existence and uniqueness of solutions  
     Properties of nonlinear vs. linear equations
     Qualitative methods for autonomous equations
     Plotting direction fields using a MSS
     Models and applications
Numerical Methods
     Introduction to a numerical solver in a MSS
     Elementary numerical methods: Euler, Improved Euler, Runge-Kutta
     Local and global error, reliability of numerical methods
Higher Order Linear Equations
     General Theory of linear equations
     Homogeneous linear equations with constant coefficients
     Reduction of order
     Methods of undetermined coefficients and variation of parameters for non-homogeneous equations
     Symbolic and numerical solutions using a MSS
     Mechanical and electrical vibrations
Laplace Transforms
     Definition and calculation of transforms
     Applications to differential equations with discontinuous forcing functions
First Order Linear Systems
     General theory
     Eigenvalue-eigenvector method for systems with constant coefficients
     Finding eigenpairs and solving linear systems with a MSS
     The phase plane and parametric plotting with a MSS
First Order Systems in the Plane
     Autonomous systems and critical points
     Stability and phase plane analysis of almost linear systems
     Linearized stability analysis and plotting vector fields using a MSS
     Numerical solutions and phase portraits of nonlinear systems using a MSS
     Models and applications