STANDARD COMPETENCIES:

 

 I.      Classify differential equations

 II.     Demonstrate rudimentary knowledge of what solutions to initial value problems are.  This includes geometric and numeric estimations of solutions.

 III.    Demonstrate knowledge of a variety of analytical techniques used to find solutions to first order differential equations.

 IV.     Use a variety of mathematical models and numerical methods to analysis applications that involve first order equations.

 V.      Demonstrate knowledge of the solution techniques of higher-order, linear differential equations.

 VI.     Use a variety of mathematical models and numerical methods to analysis applications that involve second order linear equations.

 VII.    Demonstrate knowledge of Laplace Transforms and/or Power Series.

 VIII.   Demonstrate rudimentary knowledge of linear systems.

 TOPICAL OUTLINE:

 

 I.      Classify differential equations and recognize solutions to differential equations.

         A.      Categorize differential equations (e.g. order, linear, ordinary, independent variable, etc.).

         B.      Verify explicit and implicit solutions to differential equations and initial value problems.

         C.      Discuss Existence and Uniqueness Theorem.

         D.      Use and apply directions fields.

         E.      Use and apply phase line (optional).

         F.      Use and apply Euler¿s approximation.

 II.     Develop techniques of finding solutions to first order differential equations and initial value problems.

         A.      Use the technique of separation of variables.

         B.      Solve first order linear equations.

         C.      Solve first order exact equations.

         D.      Solve first order equations using special integrating factors or substitution techniques (e.g. homogeneous, Bernoulli, linear coefficients, etc.). ¿ Optional.

 III.    Apply techniques of previous two sections to help solve applications that can be modeled with first order ordinary differential equations.

         A.      Discuss mathematical modeling and compartmental analysis.

         B.      Examine various applications including some of the following:

                 1.      Growth and Decay

                 2.      Newtonian Mechanics

                 3.      Mixture

                 4.      Heating and Cooling

         C.      Examine further numeric techniques ¿ Optional.

                 1.      Improved Euler¿s

                 2.      Runge-Kutta

 IV.     Develop techniques for solving higher-order, linear differential equations.

         A.      Utilize linear operators to facilitate understanding of linear ODEs.

         B.      Examine fundamental solution sets of Homogeneous Equations, linear independence, and the Wronskian.

         C.      Given one solution, solve for a second independent solution using reduction of order.

         D.      Examine homogeneous linear equations with constant coefficients.

         E.      Discuss Principle of Superpostion.

         F.      Examine nonhomogeneous equations

                 1.      Apply Method of Undetermined Coefficients.

                 2.      Apply Method of Variation of Parameters.

 V.      Introduce Linear Systems.

         A.      Examine modeling via systems.

         B.      Examine Elimination Method for systems.

 VI.     Apply techniques of previous two sections to help solve applications that can be modeled with second order ordinary differential equations.

         A.      Examine various applications including some of the following:

                 1.      Mass-spring oscillators and mechanical vibrations

                 2.      Interconnected Fluid Tanks

                 3.      Coupled Mass-spring systems

                 4.      Electrical Circuits

         B.      Introduce the Phase Plane (Optional)

 VII.    Develop the techniques of Laplace Transforms.

         A.      Know the definition and properties of Laplace Transforms.

         B.      Use tables to find Laplace Transform.

         C.      Use Inverse Laplace transforms and understand their properties.

         D.      Solve initial value problems using Laplace Transforms.

         E.      Examine Transforms of discontinuous and periodic functions (optional).

         F.      Examine Convolution Theorem (optional).

 VIII.   Develop the techniques of Power Series solutions.

         A.      Review Power Series and Analytic Functions.

         B.      Examine ordinary and singular points.

         C.      Find solutions near and ordinary point.