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Differential Equations (92.236)
Review Guide for Exam #3

Exam #3 will include material from Chapter 3 Sections 3.4 - 3.8, Chapter 4 Sections 4.1 and 4.3, and the first part of Chapter 7 of your text. 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:

From Chapter 3 (Sections 3.4 - 3.8), you should be able to
Find particular solutions to linear constant coefficient ODEs using the method of Undetermined Coefficients and the Variation of Parameter method.
Determine when the method of Undetermined Coefficients is appropriate (since this is usually the easier of the two methods).
Select a proper guess for the particular solution for use with the method of Undetermined Coefficients.
3.4 & 3.6
Be able to formulate, solve, and interpret solutions of ODEs that model mechanical motion of simple mass-spring-dashpot systems governed by my'' + cy' + ky = f(t).
Understand the behavior of damped and undamped systems.
Identify and solve the equations associated with forced and unforced systems.
Put solutions in the form y(t) = C cos(wt - a), if appropriate.
For periodic forcing functions, extract the transient and steady periodic solutions of damped systems.
Be able to formulate, solve, and interpret solutions of ODEs that model the dynamics of simple RLC electrical circuits.
Be able to distinguish between IVPs and BVPs (worked examples and applications of BVPs were not treated and will not be required for the Exam).

From Chapter 4 (Sections 4.1 and 4.3), you should be able to
Write a system of first order ODEs in standard vector form,
dz/dt = f(z,t)).
Convert an nth order ODE into a series of n 1st order ODEs.
Recognize that the same numerical techniques used in Chapter 2 apply to systems of 1st order ODEs written in standard vector form.
Solve vector IVPs using Matlab's ODE solver (or the routines developed earlier in the semester using the Euler, Modified Euler, or 4th order Runge Kutta methods).

From Chapter 7 (Sections 7.1 - 7.3), you should be able to
Utilize the basic definition of Laplace transforms to find F(s) given some function f(t). 
Understand and utilize a variety of Laplace transform properties (linearity, differentiation of f(t), integration of f(t), etc.). 
Find inverse Laplace transforms using simple Table Lookup. 
Solve linear constant coefficient IVPs using Laplace transforms. 
Solve a low-order system of linear constant coefficient IVPs using Laplace transforms. 
(for Final)
Use a variety of partial fraction expansion techniques to find inverse Laplace transforms. 
Recognize transfer functions with quadratic factors and simplify these by completing the square, where s2 + 2as + a2 + b2 = (s + a)2 + b2. 
Use Matlab's symbolic capability to find Laplace and inverse Laplace transforms on the computer. 
Some general familiarity with a variety of symbolic operators in Matlab is expected. In particular, you should be able to use diff and int to differentiate and integrate symbolic functions, and use solve and dsolve to symbolically solve algebraic equations and ODEs, respectively.

In addition to the specific techniques, applications, and terminology noted above, you are also responsible for the material covered earlier in this course. The focus on Exam #3 will clearly be related to the material identified above, but you certainly are expected to be able to use the techniques from Chapters 1 and 2 and the initial sections of Chapter 3, and to solve application-oriented problems that were discussed previously. For this exam, however, the applications will highlight the material from Sections 3.4 and 3.6-3.7. From the description of a physical system related to a simple mechanical system or to a simple RLC electric circuit, 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. A handout, Summary Information for the Entire Course, has been made available for your use during Exam #3 and for the Final Exam. Since Exam # 3 will be a Take Home Exam, you can also use your textbook and notes for guidance while doing this exam.

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

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