MA 354

MA 354 - Differential Equations

  1. Introduction to Differential Equations
    a) Modeling via Differential Equations: Population Growth
    i) Unlimited Population Growth
    ii) Logistic Population Models
    iii) Predator - Prey Systems
    b) Analytic, Qualitative, and Numerical Approaches

  2. First Order Differential Equations
    a) Analytic Techniques
    i) Separation of Variables
    ii) Linear Differential Equations
    iii) Change of Variables (optional)
    b) Qualitative Techniques: Slope fields
    c) Numerical Techniques: Euler's Method
    d) Extstence and Uniqueness of Solutions
    e) Bifurcations

  3. First Order Systems
    a) Modeling via Systems
    b) The Geometry of Systems
    c) Analytic Methods for Special Systems
    d) Euler's Method for Systems

  4. Linear Systems
    a) Properties of Linear Systems and the Linearity Principle
    b) Straight Line Solutions
    c) Phase Planes for Linear Systems with Real Eigenvalues
    d) Complex Eigenvalues
    e) Special Cases: Repeated and Zero Eigenvalues
    f) Second order Linear Equations
    g) The Trace-Determinant Plane

  5. Forcing and Resonance
    a) Forced Harmonic Oscillators
    b) Sinusoidal Forcing
    c) Undamped Forcing and Resonance
    d) Amplitude and Phase of the Steady State
    e) The Tacoma Narrows Bridge (optional)

  6. Laplace Transform
    a) Laplace Transforms
    b) Discontinuous Functions
    c) Second Order Equations
    d) Delta Functions and Impulse Forcing
    e) Convolutions

  7. Numerical Methods (optional as a substitution for parts of Chapter V and Chapter VI)
    a) Numerical Error in Euler's Method
    b) Improving Euler's Method
    c) The Runge-Kutta Method

Learning Outcome 1: The student will demonstrate an understanding of finding solutions to linear first order differential equations explicitly.

Learning Outcome 2: The student will demonstrate an understanding of solving higher order linear equations explicitly.

Learning Outcome 3: The student will understand power series methods and approximation methods to solve linear equatons and matrix methods to solve linear systems.