MAT 267 - Elementary Differential Equations with Laplace Transforms

• 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.