http://interval.louisiana.edu/courses/350/spring-2000-math-350_exam_hints.html

Math. 350-01, Spring, 2000 Hints for the Exams

Instructor: R. Baker Kearfott, Department of Mathematics, University of Louisiana at Lafayette
Office hours and telephone, Email: rbk@louisiana.edu.

/ The first exam / The second exam / The third 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, open-notes, computer-on.  You should know how to do the following.

• Solve a first-order linear initial value problem by hand by computing an integrating factor.
• Solve a first-order equation by hand with separation of variables.
• Use Mathematica to plot a family of solutions to a differential equation corresponding to different initial values.
• Be able to print your Mathematica notebooks.
• Know how to compare your manually derived solutions to solutions obtained with Mathematica.  (Be prepared to write down a short discussion paragraph.)
Postscript copy of the exam

The second exam:
The exam will be open-book, open-notes, computer-on. Be sure to bring your student ID card, since I (Prof. Kearfott) will not be proctoring the exam.

• You will determine the equilibrium solutions of a first-order differential equation.  You will use PlotVectorField with ScaleFunction ->(1&) and AspectRatio -> 1 to plot the vector field for the differential equation, to explain which of the equilibrium solutions are stable and which are unstable.
• You will need to solve two differential equations either by hand or using DSolve, then plot the two solutions. These two differential equations are mathematical models of a somewhat familiar phenomenon. You will need to discuss, based on the graphs of the solutions, which model you think is most appropriate.
Postscript copy of the second exam

The third exam:
The exam will be open-book, open-notes, computer-on. Be sure to prepare well, since you will be short of time otherwise.

• You will need to solve by hand a homogeneous second-order constant coefficient ODE and a non-homogeneous second-order constant coefficient ODE.  You should be familiar with how to handle multiple roots of the characteristic equation. You will check your solutions with Mathematica.

• Given physical data, you will write down the differential equation governing a damped spring system.  Be sure you understand the physical units, both in the English system and the metric system.  You will need to state the critical damping value, as well as the frequency of vibration.  You will use Mathematica to produce graphs of the solution, for various values of the damping parameter.
Postscript copy of the third exam

The final exam:
As with the exams during the semester, the exam will be open-book, open-notes, computer-on.  Here is a study guide.

• Be able to solve a simple initial value problem with Laplace transforms.  You may use the table in the book, or you may use Mathematica's <<Calculus`LaplaceTransform` with LaplaceTransform and InverseLaplaceTransform.  Be sure you write down all of your steps, or hand in your notebook. You will also need to solve the problem by another method, and show that the answers that you get are equivalent.
• Be able to compute the first few terms of the Taylor series to the solution of an initial value problem that does not have constant coefficients.
• Be able to solve a higher-order (that is, higher order than 2) constant coefficient differential equation by writing down the characteristic equation and by using undetermined coefficients.  Also, be able to show that your solution is equivalent to a solution that you obtain another way.
• Be able to solve an initial value problem for a differential equation that is an Euler equation, or else for a differential equation that has a regular singular point.  Understand that the solutions to such problems are not necessarily unique.
Postscript copy of the final exam