Final Exam Review Sheet--Math 308-511

General Information

The test will be held on Friday, May 7 from 12:30 to 2:30 pm in our usual room. It will cover the following material: from Nagle and Saff, Chapter 1; Chapter 2, through §2.3; Chapter 3, through §3.2; and Chapter 4, §4.1-4.9, and §4.10 (pp. 207-208); Chapter 5, §5.1-5.3; Chapter 7, §7.1-7.8; Chapter 9, §9.1-9.5. from Pratap (MATLAB), Chapter 1, Chapter 2, Chapter 3, §3.1-3.3, 3.5, and Chapter 5, §5.1.1, §5.1.3, and §5.5. You will be allowed to use MATLAB during the test, but not MAPLE or any program or calculator that can do symbolic manipulation. You will be provided with a (small) table of integrals and a table of Laplace transforms. The test will have 8 to 10 questions, some with multiple parts. One of the questions will require you to use MATLAB, but you will not have to print or save any files. In other questions, MATLAB will be a help, but will not be necessary for solving the problem. Please bring two 8½x11 bluebooks. For the material before the second exam, see the review sheets for Tests I and II. Below is an outline of the material covered after Test II, which will account for approximately 25% of the exam.

Linear Systems

Foundations

Normal form. Let A(t) be an n x n matrix and let f(t) be an n x 1 column vector. Entries in both of these may depend on t. The normal form of a first order linear system is
dx/dt = A(t)x+f(t).
The associated initial value problem (IVP) is the system above plus the initial conditions x(t₀)=x₀
Homogeneous equations. To find he general solution x to the homogeneous equation dx/dt = A(t)x, find a fundamental set of n linearly independent solutions, {x₁, ..., x_n}. The general solution x is then a linear combination of solutions from the fundamental set:
x = c₁ x₁(t) + c₂x₂(t) + ... + c_nx_n(t)
Using the fundamental matrix X(t)=[x₁(t) ... x_n(t)], the general solution is x(t) = X(t)c. Here, the vector c is an n x 1 column of constants, the c_k's. To solve the IVP, determine the scalars via
c=X(t₀)^-1x₀
Nonhomogeneous equations. The solution to dx/dt = A(t)x+f(t) has the form x=x_h + x_p, where x_h is a solution to the homogeneous equation and x_p is a particular solution. Finding particular solutions can be done via variation of paraments, undetermined coefficients, or even Laplace transforms.

Methods & Techniques

Solving homogeneous constant coefficient systems. To solve such systems, we proceed as follows.

Find the eigenvalues and eigenvectors of the matrix A. (See §9.4.)

If one has a linearly independent set of n eigenvectors {u₁, ..., u_n} corresponding to eigenvalues r₁,...,r_n, then the set with x_j(t)=exp(r_jt)u_j, j=1...n, is fundamental. This holds whether the eigenvalues are real or complex, and whether or not the eigenvalues are distinct. If the eigenvalues are distinct, then one automatically has a set of linearly independent eigenvectors.

If an eigenvalue is complex, so that r₊=a+ib, where a and b are real numbers, then there will be a second eigenvalue r_-=a-ib. Each contributes an eigevector, u₊=v+iw, and u_-=v-iw. One can use these to get two linearly independent complex solutions of the type described above in (2), or one can replace these with two real solutions (see §9.5):
x₁(t)=e^atcos(bt)v - e^atsin(bt)w
x₂(t)=e^atsin(bt)v + e^atcos(bt)w.

Applications

Mixing problems
Coupled spring problems

Matlab

poly, roots, eig
inv, det
residue