Spring 2008 Study Guide for the final exam 2250-1 and 2250-2


The 2250 final exams consist of twelve to fifteen problems. You are
expected to complete two or more per chapter for full credit. The
problems are divided by chapters. Only chapters 3, 4, 5, 6, 7, 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 7 Laplace rules through the first shifting 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. Matrices. Vectors. Vector space toolkit. Inverses.
            Adjugate formula. Cayley-Hamilton theorem. Determinants.
            Rank, nullity. Free and lead variables. The three
            possibilities. Basis of solutions.

 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.
            Rank. Nullity. Transpose. Theorems 1 and 2 of 4.2. Row and
            column spaces. Pivot theorem. Equivalence of 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
            rule. 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 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. Extras: Integral
             theorem. Periodic function theorem. Convolution theorem.
             Resolvent methods for x'=Ax and x''=Ax. Transfer function. 
             Unit step, square wave, sawtooth, staircase, ramp.

Final exams for 2250 with solution keys for 2004 to 2007 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.