advertisement

MATH 2280-002 Final Exam Review Math 2280 Section 002 [SPRING 2013] The following review should give you a very good idea of what the final will look like. There will be 5 questions on the exam. The skills and knowledge tested by each question are described below. Note that you may not need all of the skills/knowledge I’ve listed to complete a problem, but if you study everything this review recommends, you should be adequately prepared for final. I recommend finding problems that sound similar in the textbook (and lecture notes) and working through them in study groups or by yourself. (1) We have seen several important approaches to solving differential equations – in particular, separation of variables; integration factors; undetermined coefficients; variation of parameters; Laplace transforms; and Fourier series techniques. Of course, these techniques don’t all apply to the same problems. I will ask you to solve a linear differential equation using two different techniques. You will not be required to know variation of parameters. (2) There will be a problem asking you to solve a system of linear DE’s of the form x′ = Ax. You may be asked to find a fundamental matrix or eAt . Remember how to use the solutions to a linear system to write down either of these matrices, and remember how to use these matrices to write down the solution to an IVP using the initial conditions. Be able to handle both real and complex, distinct and repeated eigenvalues. Know how to find a chain of generalized eigenvectors so that you can deal with defective eigenvalues. (3) There will be one question on Laplace transforms. The problem will be something similar to one of the homework problems from Chapter 7. You should have the definition of the convolution memorized, but I will provide a formula sheet for Laplace transforms that includes the important properties we’ve discussed. (4) There will be a boundary value problem similar to the heated rod examples worked out during lectures. (5) I will give you one problem modeling either nonlinear mechanical vibrations or the interaction between populations. In either case, you should be able to find critical points, compute the Jacobian, give the linearization at each critical point, and use the eigenvalues of the Jacobian to classify the type and stability of critical points. I will provide you with a table with eigenvalues and the types of critical points. This table will not tell you about stability, so you will have to remember, for example, that a sink is (asymptotically) stable and a source is unstable. For a given linearization, you should be able to sketch the direction field near the critical point (0,0) just like we did in class (by making observations about when x′ is positive, negative or zero and when y ′ is positive, negative, or zero). Given the model describing the interaction of two populations, you should be able to tell me whether each population exhibits exponential or logistic growth in the absence of the other 1 MATH 2280-002 Final Exam Review population. You should also be able to tell me whether the interaction between populations is competition, cooperation, or predation. Finally, you should be able to describe the behaviour of the populations near a critical point as t → ∞. Given the model for a nonlinear mass-spring system, you should be able to tell be me whether the motion is damped or undamped and whether the spring is hard or soft. You will need to write the second-order DE I give you as a two-dimensional first-order system. I may give you initial conditions and then ask you about the behaviour near a critical point as t → ∞. 2