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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 1storder
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 PredictorCorrector Methods. 
Discuss
the relationship between error and step size for the 2ndorder
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 4thorder
RungeKutta Method. 
Perform
the steps associated with the 4thorder RungeKutta 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 builtin
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 firstorder 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 accelerationvelocity
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 realvalued
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 applicationoriented
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 accelerationvelocity
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
 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.
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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