MATH 2280-002
Exam 1 Review
Math 2280 Section 002 [SPRING 2013]
The following review should give you a very good idea of what Exam 1 will look like. There will be 10
questions on the exam. The skills and knowledge tested by each question are described below. Note
that you may not need all of the skills/knowledge I’ve listed to complete a problem, but if you study
everything this review recommends, you should be adequately prepared for Exam 1. I recommend
finding problems that sound similar in the textbook (and lecture notes) and working through them
in study groups or by yourself.
(1)-(5) There will be five multiple choice/fill-in-the-blank questions. Unlike the rest of the exam, no
partial credit will be awarded for any question in this section. You will need to know the
definitions of separable, linear, and autonomous differential equations. If I give you a DE, you
should be able to tell me which of these three descriptions apply (more than one may apply).
Know the definition of the Wronskian and know what it tells you about the solutions to a DE.
Be able to match a slope field with its differential equation. Know all of the existence and
uniqueness theorems (there’s more than one). If I give you a DE, be able to tell me if a
solution must exist and whether or not it must be unique.
(6) There will be a population problem involving an autonomous, separable DE. Be able to solve
this DE (you may need to find a partial fractions decomposition). Be able to find the critical
points, draw the phase diagram, and classify points as stable/unstable/semistable. Be able to
describe the behavior as t → ∞ based on some initial condition.
(7) There will be a mixing tank problem involving a single tank (not a cascade of tanks). Be able
to set up the differential equation on your own if you are provided with the salt concentration
of the incoming brine, the rate of the inflow and outflow of brine, the initial salt concentration
of the tank, and the initial volume of the tank. Know how to solve the resulting DE (using
integrating factors).
(8) There will be a mechanical vibration problem. It will be similar to problems #15-21 in section
3.4 (page 197) of your textbook, so know how to do these problems. I may ask you to sketch
the graph of the solution curve. In this case, I’ll only expect a rough graph – one that you should
be able to provide without the use of a computer/calculator.
(9) There will be a nonhomogeneous, linear DE with constant coefficients that I’ll ask you to
solve either using undetermined coefficients or variation of parameters. (Of course, I may
not specifiy a method and leave the choice up to you.) I may give you inital conditions and ask
you to solve the appropriate IVP.
(10) There will be one extra credit problem drawn from lecture material. (No hints here!)
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