Fall 2008 Study Guide
          Final Exam 2250-2 at 7:30 in WEB 103 on Dec 15, 2008


The 2250 final exam consists of at least sixteen problems. The problems
are divided by chapters. You are expected to complete one or two per
chapter for full credit. Only chapters 3, 4, 5, 6, 7, 8, 9, 10 appear on
the exam.

The following problems will be used as models for the problems that will
appear on the final exam. Each problem will have one to five parts, to
facilitate division of credit for that problem.

Topics outside the subject matter of these problems will not be tested.
However, theoretical questions about the details of the problem may be
asked. Generally, proofs of textbook theorems are not part of the final
exam. There is no numerical or maple work on the final exam, nor are you
asked to know anything other than basic integral tables and
differentiation formulas. The basic Laplace table (4 items) is assumed
plus the 10 Laplace rules through the convolution theorem.

 Chapter 3: 3.1-16, 3.2-18, 3.2-28, 3.3-18, 3.4-22, 3.4-29, 3.5-21,
            3.6-17, 3.6-32, 3.6-39, 3.6-60
            Frame sequence to rref. General solution. Reduced echelon
            system. Free and lead variables. The three possibilities. 
            Matrices. Vectors. Inverses. Rank, nullity. Basis of
            solutions. Elementary matrices Determinants. Adjugate
            formula. Cayley-Hamilton theorem.
           
            .

 Chapter 4: 4.1-16, 4,1-21, 4.1-31, 4.1-34, 4.2-11, 4.2-13, 4.2-19,
            4.3-17, 4.3-23, 4.4-9, 4.4-19, 4.5-9, 4.5-22, 4.6-4,
            4.7-7, 4.7-11, 4.7-21 
            Vector spaces. Subspaces. Basis. Dimension. Orthogonality.
            Vector space toolkit. Rank. Nullity. Transpose. Theorems 1
            and 2 of 4.2. Independence tests: Rank test, Determinant
            test, Sampling test, Wronskian test. Pivot theorem.
            Equivalence of bases. Subspace proofs. Finding bases.

 Chapter 5: 5.1-33 to 5.1-42, 5.3-15, 5.2-21, 5.3-1 to 5.3-20, 5.3-28,
            5.3-33 5.4-17, 5.5-4, 5.5-27, 5.5-39, 5.5-49, 5.6-9,
            5.6-13, 5.6-17, 5.6-27
            Recipe. Atoms. General solution from an atom list.
            Over-damped, critically damped, under-damped.
            Phase-amplitude solution. Undetermined coefficients. Fixup
            rules. Corrected trial solution. Variation of parameters.
            Steady state periodic solution. Pure and practical
            resonance.

 Chapter 6: 6.1-5, 6.1-13, 6.1-23, 6.1-33 to 6.1-36, 6.2-11, 6.2-17,
            6.2-25, 6.2-31 to 6.2-37
            Fourier's model. Eigenpairs. Eigenpair packages P and D.
            Complex eigenvalues and eigenvectors. Diagonalization theory
            AP=PD. Independence of eigenvectors. Similar matrices.

 Chapter 7: 7.1-19, 7.1-24, 7.2-15, 7.3-11, 7.3-17, 7.3-27, 7.3-39
            Brine tank. Railroad cars. x'=Ax for 2x2, 3x3, 4x4. Ch1+ch5
            methods. Eigenanalysis method. x''=Ax for 2x2, 3x3. The four
            methods: (1) First-order method for diagonal A. (2)
            Second-order method for non-diagonal 2x2 A. (3) The
            Eigenanalysis method. (4) Laplace resolvent method.

 Chapter 8:  8.1-4, 8.1-12, 8.1-38, 8.2-4, 8.2-19
             Fundamental matrix. Matrix exponential. Nilpotent matrix.
             Undetermined coefficients. Variation of parameters. 
             Cayley-Hamilton. Putzer's method [historical only]. Laplace
             resolvent. Transfer function.

 Chapter 9: 9.1-8, 9.1-18, 9.2-2, 9.2-12, 9.2-22, 9.3-28, 9.4-8
             Stability. Autonomous system. Direction field. Phase plane.
             Critical point. Unstable. Asymptotically stable. Attractor.
             Repeller. Spiral. Saddle. Node. Center. Linearization.
             Jacobian. Classification of almost linear systems.
             Predator-prey systems. Competing species. Co-existence.
             Oscillating populations. Competition. Inhibition.
             Cooperation. Predation. Hard spring. Soft spring. Damped
             nonlinear vibrations. Nonlinear pendulum. Undamped
             pendulum. Damped pendulum.

 Chapter 10: 10.1-11 to 10.1-32, 10.2-5, 10.2-11, 10.2-17 to 10.2-24,
             10.3-9, 10.3-19, 10.3-33, 10.3-37, 10.4-17, 10.4-18
             Shift, parts, s-diff, Lerch required plus 5-line Table.
             Solve y''=10. Solve y'-y=5-2t. Solve 2x2 system. Solve
             x''=10, y''=y'+x. Forward table methods. Partial fractions.
             Backward table methods. Integral theorem. Periodic function
             theorem. Convolution theorem. Resolvent methods for x'=Ax
             and x''=Ax. Transfer function. Unit step, square wave,
             sawtooth, staircase, ramp. Maple linear system package.

Final exams for 2250 with solution keys for 2004 to 2008 appear on the
web page

           http://www.math.utah.edu/~gustafso/index2250.html

These exams may be printed and used as a study guide. Other exams from
2006-2007 (1,2,3) are also useful as a study guide, using the above list
of problems to filter out likely problem types. Finally, the three
midterms from this semester are particularly relevant and all problem
types that have appeared already are likely to appear on the final exam.

Chapters 8, 9 are new in the syllabus, so there are no past exam
questions to study. Sample problems are listed above and exam questions
will use the same problem type.