MAT-285 Objectives

MAT-285  Differential Equations

Unit 1

  • Identify differential equations as ordinary/partial, linear/nonlinear and by order.
  • Define the solution of an ordinary differential equation (ODE).
  • Verify the solution of an ODE.
  • Identify explicit and implicit solutions to an ODE.
  • Define an n-parameter family of solutions to an ODE.
  • Define particular and singular solutions to an ODE.
  • Define an integral curve. • Define initial value problems (IVP).
  • Apply Existence of a unique Solution-Theorem for first order ODE.
  • Find differential equations that model the following problems; radioactive decay, Newton’s law of cooling, tank mixing and draining, series circuits.

Unit 2

  • Use computer software to obtain a direction field.
  • Use direction fields to sketch a solution to an IVP.
  • Identify autonomous and non-autonomous ODEs.
  • Find critical, equilibrium and stationary points to an autonomous ODE.
  • Identify asymptotically stable, unstable and semi-stable critical points.
  • Find phase portraits to an autonomous ODE.
  • Solve a first order separable ODE. • Solve a first order linear ODE.
  • Define homogeneous and non-homogeneous linear ODEs.
  • Utilize the error function as a solution to a first order linear ODE.
  • Use computer software to evaluate the error function.
  • Solve exact ODEs. • Use integrating factors to produce exact ODEs
  • Solve homogeneous ODE by substitution. • Solve a Bernoulli Differential equation.
  • Use Euler’s method to approximate the solution of an IVP.
  • Use computer software to facilitate computation of Euler’s method.
  • Calculate absolute and relative error.

Unit 3

  • Solve linear models for the following problems; bacterial growth, half-life of a chemical substance, Newton’s law of cooling, mixture and series circuit problems.
  • Solve variations of the logistic equation.
  • Find a system of differential equations that model predator-prey, series circuit and mixing applications.

Unit 4

  • Apply the Existence of a Unique Solution Theorem for an nth order linear IVP.
  • Contrast Initial Value Problems (IVP) with Boundary Value Problems (BVP).
  • Define a system of linearly independent functions on an interval.
  • Understand the form of the general solution to a linear ODE.
  • Use reduction of order techniques to solve second order linear homogeneous ODEs.
  • Solve linear homogeneous ODEs with constant coefficients.
  • Apply the method of undetermined coefficients to find a particular solution to a non-homogeneous ODE.
  • Use the method undetermined coefficients-annihilator approach to find a particular solution to a non-homogeneous ODE.
  • Use variation of parameter technique to solve second order linear ODEs.
  • Solve Cauchy-Euler equations by variation of parameter and undetermined Coefficient techniques.
  • Use computer software to solve linear ODEs.
  • Solve systems of linear ODEs by elimination.
  • Use computer software to solve a system of linear ODEs.
  • Solve non-linear ODEs with either independent or independent variable missing.
  • Find the Taylor series solution to a nonlinear ODE.

Unit 5

  • • Solve higher order linear applications in spring/mass systems and series circuit analogue.
  • Solve applied non-linear models.

Unit 6

  • Apply the Existence of a Power Series Solution Theorem for second order linear ODEs.
  • Solve second order linear ODEs by Power Series. • Apply Frobenius’ theorem to second order linear ODEs.
  • Solve applications of Bessel’s and Legendre’s equations.

Unit 7

  • Apply the definition of Laplace transform.
  • Apply inverse Laplace transforms.
  • Solve IVP and BVP using Laplace transforms.
  • Use derivatives of transforms to solve ODEs.
  • Use convolution theorem to solve ODEs.
  • Solve Volterra integral equations using Laplace transforms.
  • Use computer software to compute Laplace and inverse Laplace transforms.
  • Find the Laplace transform of periodic functions. • Find the Laplace transform of the Dirac Delta function.
  • Use Laplace transforms to solve a system of ODEs.

Unit 8

  • Use computer software to review basic operations of linear systems.
  • Apply Existence of a Unique Solution theorem for a first order linear system (FOLS).
  • Use the Wronskian to test independence of solution vectors to a FOLS.
  • Understand the form of the general solution to a FOLS.
  • Use computer software to calculate eigenvalues.
  • Use eigenvalues to solve a FOLS.
  • Use method of undetermined coefficients to solve a FOLS.
  • Use method of variation of parameters to solve a FOLS.