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Differential Equations (92.236)
Review Guide for Exam #2

Exam #2 will include all the material from Chapter 2 of your text and the first three sections from Chapter 3. Each section emphasizes a particular method, important terminology, or some typical applications of the methods discussed. In particular, the necessary skills that you should have upon completion of this portion of your text are summarized as follows:

 Section From Chapter 2, you should be able to 2.4 Explain the basic principles and develop the recurrence formulas for the Euler Method. Discuss the relationship between error and step size for the 1st-order Euler Method. Perform the steps associated with the Euler Method to approximate the solution to a given IVP with a fixed step size h (by hand using a calculator and automatically using the euler.m function file in Matlab). 2.5 Explain the basic principles and develop the recurrence formulas for the Modified or Improved Euler Method. Discuss the basic concept of Predictor-Corrector Methods. Discuss the relationship between error and step size for the 2nd-order Modified or Improved Euler Method. Perform the steps associated with the Modified or Improved Euler Method to approximate the solution to a given IVP with a fixed step size h (by hand using a calculator and automatically using the impeuler.m function file in Matlab). 2.6 Discuss the relationship between error and step size for the 4th-order Runge-Kutta Method. Perform the steps associated with the 4th-order Runge-Kutta Method to approximate the solution to a given IVP with a fixed step size h (automatically using the rk4.m function file in Matlab). Discuss the basic concept of Adaptive Step Control and use the built-in ode23.m function within Matlab to solve typical IVPs. 2.1 - 2.3 Formulate and solve IVPs associated with various population models (both analytically and numerically). Interpret the solutions to autonomous first-order systems in terms of the location and stability of the critical points. Draw the phase line and sketch solution curves for autonomous 1st order IVPs. Formulate and solve IVPs associated with various acceleration-velocity models (both analytically and numerically).

 Section From Chapter 3 (Sections 3.1 - 3.3), you should be able to 3.1 - 3.2 Recognize and appropriately use the terminology associated with linear ODEs (homogeneous solution, particular solution, general solution, unique solution, linear independence, Wronskian, linear superposition, constant coefficients, characteristic equation, etc.). Verify that given individual solutions satisfy the given ODE. Identify if two or more functions are linearly independent. Apply specific conditions for the problem to determine the arbitrary coefficients in the general solution to give the unique solution. Identify the key difference between IVPs and BVPs and give examples of both classes of problems. 3.3 Develop general solutions to homogeneous constant coefficient linear ODEs. Write the general solution for systems with complex roots using real-valued functions involving decaying or growing sinusoids. Use the Variation of Parameter method to determine a second linearly independent solution given one solution to a linear homogeneous ODE (used for situations with repeated roots). Find the unique solutions to linear constant coefficient homogeneous IVPs.

In addition to the specific techniques, applications, and terminology noted above from Chapter 2 and the first three sections of Chapter 3 of your text, you are also responsible for the material covered earlier in this course. The focus on Exam #2 will clearly be related to the material identified above, but you certainly are expected to be able to use the techniques from Chapter 1 and to solve application-oriented problems that were discussed previously. Several of the application examples from Chapter 1 included problems related to Newton's 2nd Law of Motion, Radioactive Decay, Compound Interest, Newton's Law of Cooling, Torricelli's Law, Mixing Problems, among others. Additional applied problems using a variety of population models and acceleration-velocity models from Chapter 2 also enhance your inventory of practical applications of modeling and simulation. In general, from a physical description of a system related to one of these example application areas, you should be able to

1. Formulate the basic balance equations appropriate for a given physical situation,
2. Solve the resultant balance equations, and
3. Analyze and interpret the solutions to the physical problems.

This latter sequence of objectives is the real goal for taking this course in Differential Equations.

PS. Note that the handout, Summary Information for Exam #2, along with a set of Integral Tables and Laplace Transforms will be available during the exam. You should be very familiar with all the material on these summary sheets.

Last updated by Prof. John R. White (January 2007)

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