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Differential
Equations (92.236)
Review Guide for Exam #1
Exam
#1 will include all the material from Chapter 1 of your text. Each
section emphasizes a particular method or possibly a few related
methods. In particular, the necessary skills that you should have
upon completion of Chapter 1 are summarized as follows:
Section

From
Chapter 1, you should be able to

1.1

Verify
that a given function is a solution to a given ODE. 
Use
initial conditions to determine the arbitrary constants within
the general solution. 
1.2

Solve
ODEs of the form y' = f(x) by simple integration. 
1.3

Sketch
solution curves for a firstorder ODE given the slope field. 
Apply
Theorem 1 to address the uniqueness and existence of solutions
to IVPs. 
1.4

Solve
separable ODEs of the form g(y)dy = f(x)dx. 
1.5

Solve
linear ODEs of the form y' + p(x)y = q(x). 
1.6

Identify
if an ODE in the form, Mdx + Ndy = 0, is exact. 
Solve
exact ODEs of the form Mdx + Ndy = 0. 
Find
integrating factors for nonexact ODEs (if appropriate). 
Use
a variety of substitution methods to convert a given
ODE into a form that is easier to solve. 
all

Classify
a given ODE into one of the groups discussed above and identify
an appropriate method for solving the given ODE. 
In
addition to the specific analytical solution techniques noted above,
several practical applications involving 1storder ODEs were integrated
throughout Chapter 1 of your text. Several additional detailed application
examples were also made available, including 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. From a physical description of a system
related to one of the 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 #1, 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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