Test 2 Review — Math 308-503 (Spring 2011)

General Information

Calculators. You may use scientific calculators to do numerical calculations — logs, exponentials, and so on. You may not use any calculator that has the capability of doing algebra or calculus, or of storing course material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Structure and Coverage There will be 4 to 6 questions, some with multiple parts. The problems will be similar to homework problems, examples done in class and worked out in the text. The material covered is from the following sections in the text: 3.4, 4.1-4.8.

Topics covered

First Order Systems: Complex Eigenvalues

• Complex numbers — Be able to manipulate complex numbers. Know how to find real and imaginary parts, the modulus, the argument (angle), and polar form for a complex number z. Be able to work with e^λt, for λ complex. (Appendix B).
• Matrices with complex eigenvalues — When matrices have complex eigenvalues, be able to find the corresponding eigenvectors. (Section 3.4)
• Systems — Be able to solve 2×2 homogeneous systems when the eigenvalues are complex. Be able to find the solution of a problem in real, rather than complex, form. (Section 3.4)

Second Order Linear ODEs

• Basis models — Know the differential equations for spring-mass systems, RLC circuit, and the simple pendulum. (Section 4.1; notes, 2/25/11).
• Theory — General linear second order equation, y′′+p(t)y′+q(t)y = g(t), homogeneous, g = 0, and nonhomogeneous, g ≠ 0, initial value problem, operator notation. (We will concentrate on second order differential equations, not the corresponding systems.) Superposition principle. Know how to prove Abel's Theorem for second order linear equations using the method given in class on 3/2/2011. Be able to find the Wronskian for two functions. Be able to test whether two homogeneous solutions form a fundamental set. Given a homogeneous solution, be able to find a second solution to form a fundamental set. (Section 4.2)
• Constant coefficient homogeneous equations — Be able to solve these using a solution of the form y = e^λt. Know how to get a second solution when there is a double root. (Use the Wronskian.) Know how to deal with complex roots and what the corresponding real solutions are. Be able to solve IVPs. (Sections 4.3 and 4.4)
• Cauchy-Euler equations — Be able to solve Cauchy-Euler equations using the trial solution y = x^r. (Sections 4.3 and 4.4)
• Free vibrations — Undamped: amplitude, phase, angular frequency, period. Damped: overdamped, critically damped, underdamped. Quasi-frequency, quasi-period. (Section 4.5)
• Nonohomogeneous second order equations — There are two main things that we covered in this section. First, the equation L[y] = g has a solution that splits into y = y_homogeneous + y_particular. This means that the problem splits into two parts: (1) find a fundamental set of homogeneous solutions; and, (2) find a particular solution. We did not treat the method of undetermined coefficients in detail, but we did deal with ay′′ by′ + cy = e^λt. (Section 4.6)
• Forced vibrations — Be able to find the frequency response function G(iω), the gain |G(iω)|, phase of the frequency response. Be able to the frequency that gives resonance. Be able to find transient and steady state solutions. (We only did the damped case.) (Section 4.7)
• Variation of parameters — Be able to solve nonhomogeneous problems using variation of parameters. You may start with either (24) or (25) on p. 287.