Mar 28, 2024  
2023-2024 General Catalog 
    
2023-2024 General Catalog
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MAT 267 - Elementary Differential Equations with Laplace Transforms


Course Department: Mathematics
Last Date of Approval: Fall 2018

4 Credits
Total Lecture Hours: 60
Total Lab Hours: 0
Total Clinical Hours: 0
Total Work-Based Experience Hours: 0

Course Description:
This course is the study of the elementary theory, solutions, and applications of ordinary differential equations, Laplace transforms, and series solutions to ordinary differential equations. This course will also help students gain mathematical literacy which will be of vital significance when making important life decisions. In addition, this course will help with any career that involves mathematics, decision making, or problem-solving.

Prerequisites/Corequisites: MAT 216 - Calculus II  or equivalent with “C” grade or better or obtain a letter of recommendation from the instructor indicating that the student may be advanced.  

Mode(s) of Instruction: Traditional / face-to-face, virtual

Credit for Prior Learning: There are no Credit for Prior Learning opportunities for this course.

Course Fees: None

Common Course Assessment(s): None

Student Learning Outcomes and Objectives:
 

  • Identify differential equations by type and solve them.
  • Use differential equations to model physical systems.
  • Demonstrate the theories of ordinary differential equations.
  • Use power series solutions to solve differential equations.
  • Solve initial value problems using the Laplace Transforms.

Unit Objectives

  •  Distinguish between an implicit and explicit solution.
  • Classify an equation as either an ordinary or partial differential equation.
  • Identify the order of the differential equation.
  • Classify an equation as either linear or nonlinear.
  • Discuss the conditions necessary for the existence of a unique solution.
  • Solve separable equations.
    • Rewrite an equation into a separable form.
  • Solve homogeneous equations.
    • Define what a homogeneous function is.
    • Demonstrate the two possible substitutions that are used to solve homogeneous equations.
  • Solve exact equations.
    • Show that an equation is exact.
  • Solve linear equations.
    • Find the integrating factor.
    • Multiple by the integrating factor.
    • Solve the resulting equation.
  • Show how directional fields can be used to determine the behavior of the solution curve.
  • Derive Euler’s Method both geometrically and analytically.
    • Apply Euler’s Method to solve differential equations.
    • Determine error in using Euler’s Method.
  • Use the Eigenvalue/Eigenvector method to find general solutions of linear first order constant coefficient systems of differential equations.
  • Find a fundamental matrix for linear first order constant coefficient systems of differential equations.
  • Use the method of variation of parameters to find a particular solution of a nonhomogeneous linear first order constant coefficient system.
  • Use orthogonal trajectories to graph the solution curve.
  • Use differential equations to solve radioactive decay and half-life problems.
  • Use differential equations to solve growth and decay problems.
  • Use differential equations to solve cooling problems.
  • Use differential equations to solve problems involving the mixing of two fluids.
  • Setup and solve a model for a spring-mass system.
  • Define initial-value and boundary-value problems.
  • Show that a solution to a differential equation exists and is unique.
  • Determine linear dependence and linear independence for a set of functions.
  • Identify homogeneous and nonhomogeneous linear differential equations.
  • Construct a general solution from a set of fundamental solutions.
  • Construct a second linearly independent solution from a known solution.
  • Solve homogeneous linear equations with constant coefficients using the characteristic equation approach.
    • Setup and solve the characteristic equation.
    • Demonstrate how the characteristic equation affects the general solution.
  • Use the variation of parameters method to solve linear differential equations.
  • Solve homogeneous linear equations with constant coefficients using the annihilator approach.
    • Compute the annihilator for a differential equation.
    • Use the annihilator to solve a differential equation.
  • Define the Cauchy-Euler equation.
  • Calculate the auxiliary equation.
  • Solve differential equations of Cauchy-Euler equation form.
  • Define ordinary points.
  • Define singular points.
  • Find the power series solutions about ordinary points.
  • Define regular singular points.
  • Define irregular singular points.
  • Use the Method of Frobenius to find power series solutions about regular singular points.
  • Define the Laplace Transform.
  • Show sufficient conditions necessary for a Laplace Transform to exist.
  • Calculate the Laplace Transform for some common functions.
  • Use the Laplace Transform to evaluate the Inverse Laplace Transform.
  • Apply the First Translation Theorem to find Inverse Laplace Transforms.
  • Apply the Second Translation Theorem to find Inverse Laplace Transforms.
  • Calculate the Laplace Transform of a derivative.
  • Use Laplace Transforms to solve differential equations with constant coefficients.



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