When and where

Postponed to Monday, December 12, in class 10:10–11:05 a.m.
LGRT 219

What

Sections 3.5–3.6  More on homogeneous linear systems & 2nd order ODEs
Sections 4.1–4.3  Nonhomogeneous linear systems & 2nd order ODEs;
               oscillations, beats, and resonance
Section 6.1      Laplace transforms: using them to solve 1st order ODEs
               with continuous forcing functions;
               calculating them from the definition as an improper integral

Resources

You will be given a copy of your text´s Table of Laplace Transforms to use during the exam. (But you still need to know how to derive formulas for Laplace transforms: see, for example, the last question in item 5 of the list of sample exams, below.)

Review

Wednesday, December 7, 7:00–8:00 p.m., LGRT 219 (with Marc)

Tools to bring

Pencils (or pens), erasers, ruler, calculator

Sample exams

1. This sample of part of an exam is from a course taught by one of the textbook´s authors, Paul Blanchard, at BU. The only relevant questions here for our Exam 3 are #3 and #4.
    
2. The next sample exam is on a page from Robert Devaney´s BU Web site. The only relevant questions here are 3–5.
    
3. Here is another sample exam from Devaney´s Web site. The only relevant questions here are 5 and 6. (This scanned page may take a while to load.)
    
4. Here is yet another sample exam from Devaney´s site. The only relevant questions are 2; 3 A and B. (This scanned page also may take a while to load.)
    
5. Another question: Use Laplace transforms to solve: y' + 2y = 4, y(0) = 1.
    
6. Here are a few more questions of a slightly different flavor:
   
   If y₁(t) and y₂(t) are solutions of a second-order non-homogeneous linear ODE, actually carry out the computations to show that y₁(t) - y₂(t) must be a solution of the associated homogeneous ODE.
   
   Write a couple of paragraphs about beats. Discuss the kind of ODE that leads to beats, what aspect of solutions actually results in beats and why mathematically that happens.
   
   Explain what it means to say, for a forced and damped harmonic oscillator, that the part of the solution due to the initial conditions is transient. Then explain the mathematical reasons this happens.
   
   Use the definition of Laplace transform in terms of integrals to derive the formula for the Laplace transform of the derivative y´(t) of a function y(t).

Caution: Just because a type of question appears in these sample exams does not necessarily mean it must appear on the actual exam; just because a type of question fails to appear in these sample exams does not necessarily mean it could not appear on the actual exam.

Other information

See the information about exams in the About pages (which reproduce what appears in the Course Description handout).

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