http://interval.louisiana.edu/courses/350/spring2001math350_exam_hints.html
Math. 35003, Spring, 2001 Hints for the Exams
Instructor: R.
Baker Kearfott, Department
of Mathematics, University of Louisiana
at Lafayette
Office hours
and telephone, Email: rbk@louisiana.edu.
Home
page for the course
This page will change throughout the semester.
/ The
first exam / The
second exam / The
third exam / The
fourth exam /The
final exam /
Note: Previously given exams are available below
in Postscript format, that can be printed with a Postscript printer.
The files can also be viewed and printed with Ghostscript
and GSview.
The first exam:
The first exam will be open book, oncomputer. You should know
how to do the following:

Use Mathematica to draw the direction field of a first order differential
equation, and know how to interpret the plot with regard to stable and
unstable equilibrium solutions.

Know how to determine equilibrium solutions and the stability thereof analytically.

Know how to distinguish the order of a differential equation, and know
how to distinguish whether a differential equation is linear or nonlinear.

Know how to solve first order linear differential equations with integrating
factors, and know how to check the solutions.
Answers,
page 1 (GIF)
Answers,
page 1b (Mathematica notebook)
Answers,
page 2 (GIF)
Answers,
page 3 (GIF)
Postscript
copy of the exam
The second exam:
This exam will also be open book, open computer.

You will need to solve an initial value problem involving a separable equation
by hand, and state the interval in which the solution is defined.
(Look at the problems on p. 45 of the book.)

You will need to set up and solve a mathematical model dealing with pollution
in a water system. You will need to understand the basics of how to derive
the differential equation, since one of the terms will differ from the
problems worked in class and in the book.

You will need to understand Euler's method. You will apply Euler's method
to a particular initial value problem with different stepsizes, and you
will need to explain the differences in the error as the stepsize decreases.
Answers,
page 1 (GIF)
Answers,
page 2 (GIF)
Answers,
page 3 (GIF)
Postscript
copy of the second exam
The third exam:
Study the following:

Solution of homogeneous second order constant coefficient equations, including

solution of the characteristic equation,

multiple roots of the characteristic equation,

complex roots of the characteristic equation.

The method of undetermined coefficients for the solution of nonhomogeneous
differential equations.

Solution of initial value problems.
Although the exam will be open book, oncomputer, you will need to solve
three initial value problems by hand.
Answers,
page 1 (GIF)
Answers,
page 2 (GIF)
Answers,
page 3 (GIF)
Postscript
copy of the third exam
Adobe
Acrobat (PDF) copy of the third exam
The fourth exam:
Pay particular attention to:

series expansion for the product of two functions. You should also
know how to write down the Taylor series of a known function.

how to compute the series expansion of the solution of an initial value
problem with an ODE with nonconstant coefficients.

how to solve Euler equations.
Answers,
page 1 (GIF)
Answers,
page 2 (GIF)
Answers,
page 3 (GIF)
Postscript
copy of the fourth exam
Adobe
Acrobat (PDF) copy of the fourth exam
The final exam:
The exam will be open book, open notes. Although some of the
solutions will be "byhand," it will be advantageous for you to be able
to check your answers with Mathematica. I thus recommend you practice
working with differential equations, series, and Laplace transforms in
Mathematica, outside of class and prior to the exam. Also, be prepared
to do the following:

Use Laplace transforms to solve an initial value problem involving a step
function.

Answer questions concerning critical values of either a resistance or the
damping factor in an electrical or a mechanical circuit. Also understand
what the quasiperiod is (and how to compute it), and what the amplitude
of an oscillation is, and how it might vary.

Be able to find the general solution to linear constantcoefficient DE's.

Be able to solve an initial value problem either by separation of variables
or by an integrating factor (if the equation is linear).

Understand how to use Mathematica's DSolve.

Be able to use series to write down the first few terms to the solution
of a nonconstant coefficient linear differential equation.
Answer
to problem 1 (GIF)
Answer
to problem 1 via Mathematica (Mathematica notebook)
Answer
to problem 2 (GIF)
Answer
to problem 3 (GIF)
Mathematica
check to problem 3 (Mathematica notebook)
Answer
to problem 4 (GIF)
Mathematica
notebood for problem 4
Postscript
copy of the final exam
Adobe
Acrobat (PDF) copy of the final exam