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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:
Section

From
Chapter 3 (Sections 3.4  3.8), you should be able to

3.5

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 massspringdashpot 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. 
3.7

Be
able to formulate, solve, and interpret solutions of ODEs that
model the dynamics of simple RLC electrical circuits. 
3.8

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). 
Section

From
Chapter 4 (Sections 4.1 and 4.3), you should be able to

4.1

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. 
4.3

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). 
Section

From
Chapter 7 (Sections 7.1  7.3), you should be able to

7.1

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. 
7.2

Solve
linear constant coefficient IVPs using Laplace transforms. 
Solve
a loworder system of linear constant coefficient IVPs using
Laplace transforms. 
7.3
(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 s^{2} + 2as + a^{2}
+ b^{2} = (s + a)^{2} + b^{2}. 
Use
Matlab's symbolic capability to find Laplace and inverse Laplace
transforms on the computer. 
NOTE:

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 applicationoriented problems
that were discussed previously. For this exam, however, the applications
will highlight the material from Sections 3.4 and 3.63.7. 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
 Formulate
the basic balance equations appropriate for a given physical situation,
 Solve
the resultant balance equations, and
 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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