Spring 2019 Study Guide
Final Exam 2280-1 (8:05am class)
Exam 7:30 to 10:00am in LCB 215 on Tuesday, April 30, 2019
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 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
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 Replacement Method of scaling eigenfunctions.
Superposition 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 may appear on the final
exam.
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