20804255Techniques in Ordinary Differential Equations
Course Information
Description
This course presents techniques for solving and approximating solutions to ordinary differential equations.  Topics will include solving first order differential equations, solving second-and higher-order linear differential equations, Laplace and Fourier transforms, systems of first order linear differential equations, numerical methods, and Sturm-Liouville Theory.
Total Credits
3
Course Competencies
  1. Examine basic terminology and properties of differential equations
    Assessment Strategies
    Oral, Written, Graphic and/or Skill Assessment
    Criteria
    Determine the order of a differential equation
    Distinguish between linear and nonlinear differential equations
    Distinguish between ordinary and partial differential equations
    Distinguish between single differential equations and systems of differential equations
    Determine whether a given function is a solution to a differential equation
    Generate a direction field for a first-order differential equation
    Use a direction field to sketch approximate solutions to a first-order differential equation

  2. Compute analytical solutions to first-order ordinary differential equations and initial value problems
    Assessment Strategies
    Oral, Written, Graphic and/or Skill Assessment
    Criteria
    Identify whether or not a first-order differential equation is linear
    Express a linear first-order equation in standard form
    Compute solutions to linear first-order equations using the method of integrating factors
    Identify whether a first-order equation is separable
    Compute solutions to separable first-order equations by separating variables
    Identify whether a first-order equation is exact
    Determine whether a first-order equation can be made exact via an appropriate integrating factor
    Compute solutions to exact first-order equations.
    Compute solutions to linear, separable, and exact initial value problems
    Use first-order differential equations to model and solve mixing problems, heating and cooling problems, motion problems, and other exponential growth and decay problems
    Compute and classify equilibrium points of autonomous first-order differential equations (optional)
    Use autonomous first-order differential equations to analyze population dynamics (optional)
    Use Euler’s Method to numerically approximate the solution to a first-order initial value problem (optional)

  3. Analyze differences between linear and nonlinear first-order ordinary differential equations
    Assessment Strategies
    Oral, Written, Graphic and/or Skill Assessment
    Criteria
    Apply the Existence and Uniqueness Theorem to determine whether a linear first-order initial value problem is guaranteed to have a unique solution
    Apply the Existence and Uniqueness Theorem to determine whether a nonlinear first-order initial value problem is guaranteed to have a unique solution
    Apply the Existence and Uniqueness theorem to predict the interval of definition of the solution to a linear or nonlinear initial value problem

  4. Apply the basic theory of solutions to second-order linear ordinary differential equations
    Assessment Strategies
    Oral, Written, Graphic and/or Skill Assessment
    Criteria
    Apply the Existence and Uniqueness Theorem to determine whether a linear second-order initial value problem is guaranteed to have a unique solution, and to predict the interval of this solution
    Apply the Principle of Superposition to express a solution to a second-order linear differential equation as a linear combination of elementary functions
    Compute the Wronskian of two differentiable functions
    Express the general solution of a second-order linear equation as a linear combination of two solutions whose Wronskian is not identically zero

  5. Compute analytical solutions to second-order, linear, homogeneous, constant-coefficient ordinary differential equations and initial value problems
    Assessment Strategies
    Oral, Written, Graphic and/or Skill Assessment
    Criteria
    Classify a second-order differential equation as “linear” or “nonlinear”, “homogeneous” or “nonhomogeneous”, and “constant coefficient” or “non-constant coefficient”
    Express a second-order linear differential equation in standard form
    Generate the characteristic equation for a second-order, linear, homogeneous, constant-coefficient differential equation
    Compute the solution to the differential equation when the corresponding characteristic equation has two distinct real roots
    Compute the solution to the differential equation when the corresponding characteristic equation has pure imaginary roots
    Compute the solution to the differential equation when the corresponding characteristic equation has complex roots
    Compute the solution to the differential equation when the corresponding characteristic equation has a repeated real root
    Compute the solution to a second-order initial value problem for any of the above cases
    Use second-order differential equations to model and solve unforced mechanical and electrical vibration problems

  6. Compute analytical solutions to second-order, linear, nonhomogeneous, constant-coefficient ordinary differential equations and initial value problems
    Assessment Strategies
    Oral, Written, Graphic and/or Skill Assessment
    Criteria
    Compute particular solutions to nonhomogeneous second-order equations using the method of undetermined coefficients
    Compute particular solutions to nonhomogeneous second-order equations using the method of variation of parameters
    Express the general solution to a nonhomogeneous linear equation as the sum of a particular solution and the general solution of the corresponding homogeneous equation
    Solve nonhomogeneous initial value problems
    Use second-order differential equations to model and solve mechanical and electrical vibration problems containing external forcing

  7. Compute solutions to differential equations using the Laplace Transform
    Assessment Strategies
    Oral, Written, Graphic and/or Skill Assessment
    Criteria
    Compute Laplace transforms of constant, power, exponential, and sine/cosine functions by applying the formal definition
    Use a table of Laplace transforms and algebraic techniques (completing the square, partial fraction decomposition) to find inverse Laplace transforms of rational expressions
    Use the Laplace transform to solve linear, constant-coefficient initial value problems
    Model discontinuous phenomena using step functions
    Solve initial value problems with discontinuous forcing using the Laplace transform
    Use the Dirac delta function to model impulsive phenomena
    Solve initial value problems with impulsive forcing using the Laplace transform
    Use the Convolution Theorem to express the solution to an initial value problem in terms of a convolution integral

  8. Compute solutions to systems of two linear first-order differential equations in two unknown functions
    Assessment Strategies
    Oral, Written, Graphic and/or Skill Assessment
    Criteria
    Express an nth-order differential equation in one unknown function as a system of n first-order equations in n unknown functions
    Express a system of n linear, first-order differential equations for n unknown functions in matrix/vector form
    Determine whether a set of n vectors is linearly independent
    Compute eigenvalues and eigenvectors of a 2x2 matrix
    Test a vector function to determine whether it satisfies a system of first-order differential equations
    Determine whether a set of vector functions constitutes a fundamental set of solutions to a system of first-order differential equations
    Solve 2x2 systems of linear, first-order, homogeneous, constant coefficient differential equations in the distinct real eigenvalue, complex eigenvalue, and repeated eigenvalue cases
    Describe the qualitative behavior of the solution to a 2x2 system of first-order differential equations by analyzing a phase portrait
    Compute a fundamental matrix for a 2x2 system of first-order differential equations (optional)

  9. Compute solutions to differential equations using series methods
    Assessment Strategies
    Oral, Written, Graphic and/or Skill Assessment
    Criteria
    Form general power series representations for a function and its first two derivatives
    Distinguish between “ordinary points”, “regular singular points”, and “irregular singular points” of linear differential equations having one or more nonconstant coefficients
    Compute the recurrence relation and corresponding series solutions near an ordinary point for a second-order differential equation
    Compute the solution to a second-order initial value problem using series methods
    Compute solutions to Euler equations
    Compute series solutions to a second-order differential equation near a regular singular point
    Apply the Taylor or Maclaurin series formula to compute several terms in a series solution to a first- or second-order initial value problem
    Solve Euler Equations (optional)
    Compute the indicial equation, the exponents of the singularity, recurrence relation, and corresponding series solutions for a second-order differential equation near a regular singular point (optional)
    Compute solutions to Bessel’s Equation of various orders (optional)