MTH 264 - Introduction to Ordinary Differential Equations

1. Develop the ability to recognize, classify, and solve different types of first-order differential equations.
   1. Identify and solve separable and linear first-order equations.
   2. Use slope fields and equilibrium solutions to understand the qualitative properties of first-order equations.
   3. Use the concept of bifurcation to understand the qualitative properties of a family of first-order equations.
   4. Apply numerical methods to generate approximations to solutions of first-order equations.
   5. Understand the conditions that guarantee the existence and uniqueness of solutions to first-order equations.
2. Solve first-order systems of differential equations and demonstrated knowledge of properties and applications.
   1. Use direction fields and equilibrium solutions to understand the qualitative properties of first-order systems.
   2. Solve decoupled and partially-decoupled systems.
   3. Apply numerical methods to generate approximations of solutions of first-order systems.
   4. Investigate the special properties of linear systems.
   5. Classify and solve first-order linear systems with constant coefficients.
   6. Use first-order linear systems to investigate the properties and solve equations arising from harmonic oscillation.
   7. Linearize nonlinear systems when appropriate.
   8. Apply appropriate quantitative, qualitative, and numerical techniques to study nonlinear systems.
3. Analyze and solve second-order differential equations and use them to various applications.
   1. . Identify homogeneous and nonhomogeneous linear differential equations.
   2. Construct particular and general solutions to homogeneous linear differential equations.
   3. Construct particular and general solutions to linear differential equations.
   4. Solve linear differential equations with constant coefficients.
   5. Use second-order linear differential equations to model damped/undamped forced/unforced oscillations.
   6. Apply power series to solve or approximate solutions of differential equations.
4. Use Laplace transforms to solve a variety of differential equations.
   1. Apply the definition and properties of the Laplace transform.
   2. Apply Laplace transforms to various fundamental functions.
   3. Apply the shifting theorems to a variety of functions and equations.
   4. Use Laplace transforms to solve a variety of initial value problems.
   5. Understand and use the Laplace transform in applications of discontinuous forcing functions.
   6. Use the convolution theorem on appropriate first- and second-order equations.
5. Use appropriate technology to investigate and solve differential equations.
   1. Generate and graph numerical solutions with a computer algebra system.
   2. Graph and recognize the relationships between forcing functions and solutions to harmonic oscillation.
   3. Recognize initial conditions in the graphs of solutions to first-order equations and systems.
   4. Generate and graph slope fields and direction fields for first-order equations and systems.
   5. Recognize and verify the correspondence between slope/direction fields and solutions of equations or systems.
   6. Graph multiple representations of solutions to first-order systems.
6. Communicate effectively about differential equations and their applications.
   1. Verbally describe solutions to problems using appropriate terminology.
   2. Provide complete written solutions to problems using appropriate terminology.
   3. Use appropriate vocabulary.