MATH 2280-002 Exam 2 Review
Math 2280 Section 002 [SPRING 2013]
The following review should give you a very good idea of what Exam 2 will look like. There will be 6 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 Exam 2. I recommend finding problems that sound similar in the textbook (and lecture notes) and working through them in study groups or by yourself.
(1) There will be a problem that asks you to compute the operational determinant of a system of linear DE’s and then use this operational determinant to solve this system of DE’s. If the operational determinant is nonzero, you should be able to tell me how many arbitrary constants appear in the general solution. If the operational determinant is 0, you should be able to tell me whether the system has no solutions or infinitely many solutions. (Take a look at problems
4.2.1-19,26-29 of your textbook.)
(2) There will be a mixing tank problem involving either a series of cascading tanks or a closed tank system. Either two or three tanks will be involved. You will need to know how to set up the corresponding system of DE’s in the form x
0
= A x and how to solve this system using the eigenvalue method. (See problems 5.2.27-37.)
(3) There will problem involving a coupled mass-spring system subject to an external, periodic force. I will set the problem up for you, but you will need to identify the natural frequencies and determine if resonance occurs. The problem will look a lot like 5.3.8-10,14-15.
(4) You will need to solve a system of the form x
0
= A x , where A is either a 2 × 2 or 3 × 3 matrix with a defective, multiplicity 2 eigenvalue, by finding chains of generalized eignevectors . The problem will look a lot like 5.4 Example 3,5.4.23-24, or one of the lecture examples from this week.
(5) You will need to solve an IVP x
0
= A x, x (0) = x
0 using a fundamental matrix for this system.
To get a fundamental matrix , you will asked to exponentiate t A .
A will either be
– a 2 × 2 or 3 × 3 diagonal matrix;
– a 2 × 2 matrix having an eigenvalue with multiplicty 2; or
– a 3 × 3 matrix having an eigenvalue with multiplicty 3.
Using this fundamental matrix, you can then compute the solution to the given initial value problem. For practice, see 5.5.25-30 and the lecture examples from this week.
(6) There will be one extra credit problem drawn from lecture material. It will probably have to do with showing that the Wronskian is zero if and only if the solutions to x
0
= A x are linearly dependent. I’ll probably ask you to identify where we need the fact that we have solutions to a system of linear DE’s and not just some functions.
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