Portland Community College | Portland, Oregon

This is a one-term introduction to ordinary differential equations with applications. Topics include classification of, and what is meant by the solution of a differential equation, first-order equations for which exact solutions are obtainable, explicit methods of solving higher-order linear differential equations, an introduction to systems of differential equations, and the Laplace transform. Applications of first-order linear differential equations and second-order linear differential equations with constant coefficients will be studied.

Upon successful completion students should be able to:

• Analyze real world scenarios to recognize when ordinary differential equations (ODEs) or systems of ODEs are appropriate, formulate problems about the scenarios, creatively model these scenarios (using technology, if appropriate) in order to solve the problems using multiple approaches, judge if the results are reasonable, and then interpret and clearly communicate the results.
• Appreciate ODE and system of ODEs concepts that are encountered in the real world, understand and be able to communicate the underlying mathematics involved to help another person gain insight into the situation.
• Work with ODEs and systems of ODEs in various situations and use correct mathematical terminology, notation, and symbolic processes in order to engage in work, study, and conversation on topics involving ODEs and systems of ODEs with colleagues in the field of mathematics, science or engineering.
• Enjoy a life enriched by exposure to Calculus.

1. Demonstrate an understanding of the theory of ODEs and their applications in:
   1. At least two proctored exams, one of which is a comprehensive final that is worth at least 25% of the overall grade; proctored exams should be worth at least 50% of the overall grade
   2. and at least one of the following:
      1. Take home exam(s)
      2. Quizzes
      3. Computer lab assignments
      4. Homework
      5. ODE related Community-Based Learning
2. Demonstrate the ability to communicate with colleagues on the topics of ODEs and systems of ODEs in:
   1. At least one group or individual project with written report and/or oral in-class presentation.
   2. and at least one of the following:
      1. Participation in in-class discussions.
      2. In-class group activities
      3. Attendance

SKILLS

1. Context Specific Skills
   • Students will learn what an ODE is, what constitutes a solution, what initial value problems are, and what constitutes a solution. They will learn to classify ODEs.
   • Students will learn to visualize and manipulate ODEs in graphical, numerical, and symbolic form.
   • Students will understand the concepts of existence and uniqueness of solutions.
   • Students will recognize certain basic types of first order ODEs for which exact solutions may be obtained and to apply the corresponding methods of solution.
   • Students will explore some of the basic theory of linear ODEs, gain ability to recognize certain basic types of higher-order linear ODEs for which exact solutions may be obtained, and to apply the corresponding methods of solution.
   • Students will be introduced to the concept of the Laplace transform and the application of the Laplace transform in the solution of constant coefficient, linear ODEs.
   • Students will be introduced to systems of ODEs and discuss graphical, numerical, and analytical solution methods.
   • Students will work with a variety of applications, using appropriate models, and will analyze the validity of the solutions obtained.
2. Learning Process Skills
   • Classroom activities will include lecture/discussion and group work.
   • Students will communicate their results in oral and written form.
   • Students will apply concepts to models of real world problems.
   • The use of calculators and/or computers for tasks such as plotting direction fields and obtaining exact and numerical solutions will be demonstrated and encouraged by the instructor where appropriate.

THEMES, CONCEPTS, ISSUES

1. DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS
   1. Identify origins and applications of differential equations.
   2. Describe what is meant by Solutions of Differential Equations.
   3. Discuss what is meant by Initial-Value Problems, Existence and Uniqueness of solutions.
   4. Use direction fields and isoclines to draw various solution curves for a differential equation.
2. FIRST ORDER EQUATIONS
   1. Qualitative Analysis of Solutions of First Order Autonomous Equations.
      1. Define and identify any equilibrium solutions.
      2. Use phase lines to classify equilibrium solutions and analyze long term behavior.
      3. Use direction fields and phase lines to sketch some solution curves.
      4. Obtain graphical solutions using Euler's method and other numerical techniques using appropriate technology.
   2. Separable Differential Equations.
      1. Use direction fields to sketch some solution curves.
      2. Solve symbolically by separation of variables and integration.
      3. Obtain graphical and exact solutions using differential equations technology.
   3. Linear Differential Equations.
      1. Use analytical methods such as Integrating Factors, Undetermined Coefficients, and Variation of Parameters.
      2. Obtain graphical and exact solutions using differential equations technology.
3. SYSTEMS OF DIFFERENTIAL EQUATIONS
   1. Define and identify equilibrium solutions for autonomous systems.
   2. Construct vector fields and use them to construct phase portraits.
   3. Use phase portraits to analyze long term behavior of solutions.
   4. Solve systems of linear differential equations analytically.
   5. Classify equilibrium points.
   6. Solve systems of first order non-linear equations with the use of technology.
4. HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS
   1. Reduce a higher order equation to a system of first order equations.
   2. Solve second order Homogeneous Equations with Constant Coefficients.
   3. Solve Non-Homogeneous Linear Differential Equations using the Method of Undetermined Coefficients.
   4. Solve Non-Homogeneous Linear Differential Equations with discontinuous forcing functions using the method of Laplace Transforms.
   5. Obtain exact and numerical solutions using differential equations technology.
5. THE LAPLACE TRANSFORM
   1. Define the Laplace transform and discuss existence and basic properties.
   2. Use inverse Laplace transform to return familiar functions.
   3. Use the Convolution theorem to work with inverse transforms of products.
   4. Use Laplace transforms to solve Linear Differential Equations with constant coefficients.
   5. Define the unit step function and the Dirac delta function. Then, use the Laplace transform to solve problems with forcing functions described in terms of step functions and impulses.
6. APPLICATIONS OF DIFFERENTIAL EQUATIONS
   1. Study at least FIVE of the following applications.
      1. Oscillations of a mass on a spring -- free, undamped and damped motion.
      2. The Logistics Equation.
      3. Series RLC circuits.
      4. Mixture problems.
      5. Deriving differential equations from measured data.
      6. Chemical reactions.
      7. Predator-Prey problems.
      8. Pursuit problems.
      9. Linearization of Non-Linear Equations about equilibrium.
      10. Van der Pol equation.
      11. Other applications of comparable depth.