http://interval.louisiana.edu/courses/350/spring-2001-math-350_exam_hints.html

### Math. 350-03, 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.

/ 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, on-computer.  You should know how to do the following:

1. 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.
2. Know how to determine equilibrium solutions and the stability thereof analytically.
3. Know how to distinguish the order of a differential equation, and know how to distinguish whether a differential equation is linear or nonlinear.
4. Know how to solve first order linear differential equations with integrating factors, and know how to check the solutions.
Postscript copy of the exam

The second exam:
This exam will also be open book, open computer.

1. 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.)
2. 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.
3. 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.
Postscript copy of the second exam

The third exam:
Study the following:

• Solution of homogeneous second order constant coefficient equations, including
1. solution of the characteristic equation,
2. multiple roots of the characteristic equation,
3. complex roots of the characteristic equation.
• The method of undetermined coefficients for the solution of non-homogeneous differential equations.
• Solution of initial value problems.
Although the exam will be open book, on-computer, you will need to solve three initial value problems by hand.
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 non-constant coefficients.
• how to solve Euler equations.
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 "by-hand," 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 quasi-period 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 constant-coefficient 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 non-constant coefficient linear differential equation.