Fall 2009 Study Guide Final Exam 2250-2 (7:30 class) at 8:00am in WEB 103 on Tuesday, Dec 15, 2009 Final Exam 2250-4 (2:25 class) at 1:00pm in JWB 335 on Thursday, Dec 17, 2009 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. Fundamental skills from chapters 1 and 2 are required. This includes the variable separable method in section 1.4, the linear integrating factor method in section 1.5, the stability definition and intuition from the scalar case in section 2.2 and the position-velocity substitution from section 2.3. 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, including the Heaviside function and the Delta function. 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 Roots. Atoms. General solution from an atom list. Over-damped, critically damped, under-damped. Phase-amplitude solution. Undetermined coefficients. Shortest trial solution. Variation of parameters. Steady state periodic solution. Pure and practical resonance. Beats. Mechanical oscillators. Electric circuits. Pendulum. Tacoma narrows bridge. London Millennium bridge. Wine glass experiment. 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 Eigenpairs. Eigenpair packages P and D. Complex eigenvalues and eigenvectors. Diagonalization theory AP=PD. Independence of eigenvectors. Similar matrices. Slides: Data conversion example. Eigenpair equations. Eigenanalysis history. Fourier's model equivalent to AP=PD. 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. Linear integrating factor method. Eigenanalysis method for x''=Ax for 2x2, 3x3. The four methods: (1) First-order method for triangular A. (2) Cayley-Hamilton Method to solve u'=Au for any square matrix A. (3) The Eigenanalysis method. (4) Laplace resolvent method for u'=Au and x''=Ax+F(t). Earthquakes. Home heating with space heater and furnace. Pollution in 3 lakes. Cascades. Recycled brine tanks. Drug elimination in the human body [mercury, lead, aspirin]. Chapter 8: 8.1-4, 8.1-12, 8.1-38, 8.2-4, 8.2-19 Method (5): u(t)=e^(Ct)u(0) solves u'=Cu. Fundamental matrix. Matrix exponential e^(At). Nilpotent matrix and e^(At) series. Methods: (1) Putzer's e^(AT) formula. (2) e^(At)=X(t)X(0)^(-1) [X=fundamental matrix]. (3) maple exponential. Undetermined coefficients. Variation of parameters. Laplace resolvent. Vector equation Transfer Function. Cayley-Hamilton Theorem as the basis for Spectral methods. Jordan form and generalized eigenvectors [enrichment]. Chapter 9: 9.1-8, 9.1-18, 9.2-2, 9.2-12, 9.2-22, 9.3-28, 9.4-8 Theory: Stability. Autonomous system. Direction field. Phase plane. Equilibria. Unstable. Asymptotically stable. Attractor. Repeller. Spiral. Saddle. Node. Center. Linearization. Jacobian. Classification of almost linear systems. Theorem 2 in 9.2. How to apply Theorem 2 when using the maple 12 phase portrait tool. Applications: 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. Maple phase portrait tool. Maple DynamicSystems package. 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. Also EPbvp7.6, delta function problems. Transfer function. Rules: Shift, parts, s-diff, Lerch. Table: 5-line brief Table. Solve y''=10. Solve y'-y=5-2t. Solve a 2x2 system. Solve a second order system x''=10, y''=y'+x. Forward table methods. Partial fractions. Backward table methods. Integral theorem. Periodic function theorem. Convolution theorem. Resolvent methods for u'=Cu and x''=Ax. Transfer function. Unit step, square wave, sawtooth, staircase, ramp. Delta function. Heaviside function. Piecewise defined functions. Solving u'=Cu by the Laplace resolvent method [(sI -C)L(u)=u(0)]. Solving x''=Ax+F(t) by the Laplace resolvent method [(s^2 I - A)L(x)=u'(0)+u(0)s+L(F)]. Maple DynamicSystems package. Maple inttrans package. Final exams for 2250 with solution keys for 2008 and 2009 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 since Fall 2008, so there are few past exam questions to study. Sample problems are listed above and exam questions will use the same problem type.