MTH 211

INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS

First-order and linear higher-order differential equations; matrices, determinants, and systems of differential equations; numerical and power series methods of solution; the Laplace transform. (Four Credits)


Course Outline (Boyce, Diprima, and Meade )

  • Introduction
    • Some Basic Mathematical Models; Direction Fields
    • Solutions of Some Differential Equations
    • Classification of Differential Equations

  • First-Order Differential Equations
    • Linear Differential Equations; Method of Integrating Factors
    • Separable Differential Equations
    • Modeling with First-Order Differential Equations
    • Differences Between Linear and Nonlinear Differential Equations
    • Autonomous Differential Equations and Population Dynamics
    • Exact Differential Equations and Integrating Factors
    • Numerical Approximations: Euler’s Method
    • The Existence and Uniqueness Theorem
    • First-Order Difference Equations

  • Second-Order Linear Differential Equations
    • Homogeneous Differential Equations with Constant Coefficients
    • Solutions of Linear Homogeneous Equations; the Wronskian
    • Complex Roots of the Characteristic Equation
    • Repeated Roots; Reduction of Order
    • Nonhomogeneous Equations; Method of Undetermined Coefficients
    • Variation of Parameters
    • Mechanical and Electrical Vibrations
    • Forced Periodic Vibrations

  • Higher-Order Linear Differential Equations
    • General Theory of n𝗍𝗁 Order Linear Differential Equations
    • Homogeneous Differential Equations with Constant Coefficients
    • The Method of Undetermined Coefficients
    • The Method of Variation of Parameters

  • Series Solutions of Second-Order Linear Equations
    • Review of Power Series
    • Series Solutions Near an Ordinary Point, Part I
    • Series Solutions Near an Ordinary Point, Part II
    • Euler Equations; Regular Singular Points
    • Series Solutions Near a Regular Singular Point, Part I
    • Series Solutions Near a Regular Singular Point, Part II
    • Bessel’s Equation

  • The Laplace Transform
    • Definition of the Laplace Transform
    • Solution of Initial Value Problems
    • Step Functions
    • Differential Equations with Discontinuous Forcing Functions
    • Impulse Functions
    • The Convolution Integral

  • Systems of First-Order Linear Equations
    • Introduction
    • Matrices
    • Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
    • Basic Theory of Systems of First-Order Linear Equations
    • Homogeneous Linear Systems with Constant Coefficients
    • Complex-Valued Eigenvalues
    • Fundamental Matrices
    • Repeated Eigenvalues
    • Nonhomogeneous Linear Systems

  • Numerical Methods
    • The Euler or Tangent Line Method
    • Improvements on the Euler Method
    • The Runge-Kutta Method
    • Multistep Methods
    • Systems of First-Order Equations
    • More on Errors; Stability

  • Nonlinear Differential Equations and Stability
    • The Phase Plane: Linear Systems
    • Autonomous Systems and Stability
    • Locally Linear Systems
    • Competing Species
    • Predator – Prey Equations
    • Liapunov’s Second Method
    • Periodic Solutions and Limit Cycles
    • Chaos and Strange Attractors: The Lorenz Equations
Contact R.L. Pryor