Spring 2018 Study Guide Final Exam 2280-2 (12:55pm class) Exam 12:45-3:15pm in LCB 219 on Friday, April 27, 2018 The 2280 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 1, 2, 3, 4, 5, 6, 7, 9 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. It is expected that you will study the solution key to the sample final exam. The sample exam has 27 problems, most with multiple parts. The actual final has 8 problems, each problem about 15 minutes total time. Topics outside the subject matter of the sample 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 derivative formulas. This includes but is not limited to the first 20 integral table entries in the textbook. The basic Laplace table (4 items) is assumed plus the 10 Laplace rules through the convolution theorem, including the unit step and the Dirac impulse. Chapter 1: 1.2-7,8,10; 1.3-15,27; 1.4-15,17,39; 1.5-5,17,23,39; Chapter 2: 2.1-7,17; 2.2-9,17; 2.3-9,23 Quadrature method, Picard theorem on existence-uniqueness, separable equation, applications of first order equations, linear first order, integrating factor method, cascade of two tanks, Verhulst logistic equation, population dynamics, stability: funnel, spout, node, phase diagram, linear drag model, nonlinear drag model, parachute problem. No numerical work will appear on the final exam, which excludes sections 2.4, 2.5, 2.6. Chapter 3: 3.1-33 to 3.1-42, 3.3-15, 3.2-21, 3.3-1 to 3.3-20, 3.3-28, 3.3-33 3.4-17, 3.5-4, 3.5-27, 3.5-39, 3.5-49, 3.6-9, 3.6-13, 3.6-17, 3.6-27, 3.7-12, 3.7-18 Roots. Euler atoms. General solution from a list of atoms. Over-damped, critically damped, under-damped. Phase-amplitude solution. Undetermined coefficients. Shortest trial solution. Variation of parameters. Steady state periodic solution. Pure and practical mechanical resonance. Beats. Mechanical oscillators. Pendulum. Tacoma narrows bridge. London Millennium bridge. Wine glass experiment. Electrical circuits, electrical resonance. Chapter 4: 4.1-6, 4.1-16, 4.1-19, 4.2-12 Chapter 5: 5.2-11, 5.2-23, 5.2-39,5.3-3, 5.3-24, 5.5-3, 5.5-13, 5.6-13 Cayley-Hamilton-Ziebur method from 4.1 examples 6,7,8. C-H-Z for solving x'=Ax for 2x2, 3x3, 4x4. Linear integrating factor method for linear cascades. Eigenanalysis method for x''=Ax for 2x2, 3x3. The four methods: (1) Linear cascade first-order method for triangular A. (2) Cayley-Hamilton-Ziebur 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). (5) Exponential matrix, fundamental matrix Putzer's formula for exp(At). Brine tanks. Vibrations 4.2. Railroad cars. Home heating with space heater and furnace. Pollution in 3 lakes. Cascades. Recycled brine tanks. Drug elimination in the human body [mercury, lead, aspirin], which appears in optional Maple Lab 10. Applications: [not on final exam] Earthquakes. Boxcars. Coupled spring-mass system modeling and symmetry. Chapter 6: 6.1-8, 6.1-18, 6.2-2, 6.2-12, 6.2-22, 6.3-28, 6.4-8 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 6.2. How to apply Theorem 2 when using the maple phase portrait tool or Rice university pplane. Applications: [not on the final exam] 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 [enrichment]. Chapter 7: 7.1-11 to 7.1-32, 7.2-5, 7.2-11, 7.2-17 to 7.2-24, 7.3-9, 7.3-19, 7.3-33, 7.3-37, 7.4-17, 7.4-18, 7.5-5, 7.5-15, 7.6-5. 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. Dirac impulse. 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 [enrichment]. Maple inttrans package. The second shifting theorem: unit step function solutions and Dirac impulse inputs. Transfer function. Applications: [not on the final exam] Hammer hits, Paul Dirac's impulse model. Home heating. Earthquakes. Boxcars. Coupled spring-mass system modeling and symmetry. Chapter 9: 9.1-5, 9.1-10, 9.1-11, 9.1-19, 9.2-3, 9.2-19, 9.3-3, 9.3-13, 9.4-1, 9.5-3, 9.6-3 Periodic functions. Even and odd functions. Periodic extension. Fourier series. Fourier coefficient formulas. Orthogonal functions. Orthogonal series. Piecewise smooth functions. Fourier convergence theorem. Gibbs over-shoot. Fourier sine and cosine series. Even and odd extensions. Integration and differentiation of Fourier series. Fourier series solutions and undetermined coefficient methods. Stedy-state periodic solutions as Fourier series. Resonance condition. Heat equation for the rod. Fourier's 1822 solution and Fourier's method of scaling eigenfunctions. Superpostion and product solutions. The method of separation of variables. Ice-pack ends. String equation and separation of variables. Shape and velocity conditions. Normal modes. Steady-state heat conduction on a plate. The midterm exams from this semester are particularly relevant and all problem types that have appeared already are likely to appear on the final exam. ===end===