Final Exam Review Sheet--Math 308-200

General Information. The test will be given on Friday, May 4, from 12:30-2:30 pm, in our usual classroom. Please bring an 8½x11 bluebook. It will cover chapter 2, §§2.4-2.6, 2.9-2.13, chapter 3, §§3.1, 3.8-13, phase plane, and series. The test will have 6 to 8 questions, some with multiple parts. Questions will resemble homework problems or examples worked in class, and the points per question will be distributed roughly according to the time spent in class on a topic. You will be given tables of Laplace transforms and of integrals, and you will be allowed qualified to use MATLAB and MAPLE, but important steps must all be justified.

Second order, linear ODEs. Here is a list of methods we discussed.

  1. Variation of parameters (§2.4 & §3.12). This is a general method for solving nonhomogeneous ODEs, both 2nd order equations and 1st order systems; it applies whether or not the coefficients involved are constant and it only requires knowing a fundamental set of solutions to the corresponding homogeneous problem.

  2. Undetermined coefficients (§2.5). This is more restrictive than variation of parameters; it applies to the constant coefficient, 2nd order ODE case in which the nonhomogeneous term is of the form polynomial × exponential × sine and cosine terms. If we use complex exponentials, then the nonhomogeneous term reduces to polynomial × complex exponential. Be able to use it to do the applications that are given in §2.6.

  3. Laplace transforms (§§2.9-2.13 & §3.13). Know the definition and be able to establish simple properties of Laplace transforms. Be able to solve problems involving step functions, impulse forces (Dirac delta functions), and convolutions. We also discussed how to use it to solve constant coefficient systems of first order ODEs. You will be alloowed to use MATLAB or MAPLE only to do inverse Laplace transforms of rational functions. You will be required to justify any further steps via properties of the Laplace transform.

  4. Power series. We covered this in class on Thursday, 27 April. See examples 1-3 in §2.8 (pp. 185-193) for solving equations similar to what was done in class. Be able to find the first few coefficients in a power series solution.

Systems of first order ODEs. Be able to convert a given differential equation into a first order system. Be able to use the eigenvalue method to solve constant coefficient systems, even in cases where the eigenvalues are complex or very simple cases of repeated eigenvalues. Know what a fundamental matrix is, and be able to find it and use it to solve initial value problems. Know the definition of the matrix exponential, eA t, and how it is related to fundamental solutions. Be able to solve simple nonhomogeneous problems via variation of parameters or by taking the Laplace transform of a system.

Phase plane. Be able to do simple examples of phase plane diagrams, including asymptotic behavior of solutions, for the systems discussed on Tuesday, 25 April.