2250 F2008 7:30 Lecture Record
Updated: Thursday December 11: 17:32PM, 2008
Today: Sunday July 15: 20:55PM, 2018
Week 15: 08 Dec, |
09 Dec, |
10 Dec, |
11 Dec, |
12 Dec, |
Week 14: 01 Dec, |
02 Dec, |
03 Dec, |
04 Dec, |
05 Dec, |
Week 13: 24 Nov, |
25 Nov, |
26 Nov, |
Week 12: 17 Nov, |
18 Nov, |
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Week 11: 10 Nov, |
11 Nov, |
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Week 10: 03 Nov, |
04 Nov, |
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Week 9: 27 Oct, |
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Week 8: 20 Oct, |
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Week 7: 06 Oct, |
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08 Oct, |
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Week 6: 29 Sep, |
30 Sep, |
01 Oct, |
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Week 5: 22 Sep, |
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Week 4: 15 Sep, |
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Week 3: 08 Sep, |
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Week 2: 02 Sep, |
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Week 1: 25 Aug, |
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28 Aug. |
29 Aug. |
Week 15, Dec 1 to 12: Sections 8.2, 9.1, 9.2, 9.3
08 Dec:
Lecture: 8.1, 8.2. Methods for solving 2x2 systems. Exponential
matrix. Laplace resolvent method for exp(At). Putzer algorithm for
exp(At). General solution u=exp(At)u0 for u'=Au, u(0)=u0. Other
ways to compute exp(At). Undetermined coefficients. Variation of parameters
example. Intro to stability theory for autonomous systems. Exercises ch8
and ch9.
09 Dec:
Lecture: Classification of equilibria for u'=Au when det(A) is not zero,
for the 2x2 case. Spiral, saddle, center, node. Linearization theory. Jacobian.
Transfer of stability: Re(lambda)<0 ==> asym. stability. Unstable solutions.
Nonlinear stability theory: when the linearized classification and stability
transfers to the nonlinear system. Exercises ch8 and ch9.
10 Dec:
Lecture: More on stability, phase diagrams, classification.
Predator-Prey systems. How to tell which is the predator and which is
the prey. Calculations for equilibrium points, linearization,
classification of equilibria, impact on the phase diagram. Some sample
code for using DEtools and DEplot in maple to make phase diagrams.
Exercises 9.1, 9.2.
11 Dec:
Final exam review started. Covered ch5 mostly, some of ch10. Packet
distributed.
Less contact with ch3, ch4, ch6 due to extra chapters 8,9 on the
final, as compared to the S2008 final exam. Not much change for ch5,
ch7, ch10. The new spin is only additional methods for solving DE,
especially exp(At) and Laplace resolvent for systems.
12 Dec:
Final exam review continued. ch10 Laplace problems. Some ch8 and ch9
problems [new material for the final exam].
Week 14, Dec 1 to 5: Sections 7.4, 8.1, 9.1, 9.2
01 Dec:
Lecture: Sample exam 3 solutions to problems 1,2. Slides on Brine tanks
and home heating.
02 Dec:
Lecture: Problems 7.3, 7.4, 8.1, 8.2. Fundamental matrix. Matrix exponential.
Various ways to solve x'=x-5y, y'=x-y, x(0)=1, y(0)=2. Eigenanalysis method.
Cayley-Hamilton hybrid method. Laplace resolvent. Exponential matrix using maple.
Projection: glass-breaking video.
More on resonance, including practical resonance theory.
Wine glass breakage (QuickTime MOV)
Wine glass experiment (12mb mpg 2min video)
Tacoma Narrows Bridge Nov 7, 1940 (18mb mpg 4min video)
03 Dec:
Lecture: More sample exam 3 solutions. Distribute sample exam 3 key.
Laplace theory problems.
04 Dec:
Exam 3 at 7am.
05 Dec:
Lecture: Systems of equations, spectral theory, variation of parameters for systems.
Topics in progress:
Main theorem (ch7) on solving u'=Au by eigenanalysis. Brine tank example
revisited.
Lecture: Second order systems. Examples are railway cars,
earthquakes, Tacoma narrows bridge, vibrations of multi- component systems.
Systems theory and examples, 800k.
(pdf manuscript)
Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation.
(pdf slides)
Review and Drill: How to
write up a solution which postpones partial fraction evaluation of
constants to the end.
Lecture: Partial fraction methods for complex roots and roots of multiplicity
higher than one.
Partial Fractions Examples: How to deal with
complex factors like s^2+4. Heaviside's coverup method and how it works in
the case of complex roots.
Week 13, Nov 24 to 26: Sections 6.2,7.1,7.2,7.3
24 Nov:
Lecture: Eigenanalysis, ch6. Calculation of eigenpairs to produce Fourier's model.
Connection between Fourier's model and a diagonalizable matrix.
How to find the variables lambda and v in Fourier's
algebraic eigenanalysis slides 2008 ( pdf)
Eigenanalysis-I manuscript S2007 (typeset 19 pages, 200k pdf)
model using determinants and frame sequences.
Solved in class: examples similar to the problems in 6.1 and 6.2.
Slides and problem notes exist for 6.1 and 6.2 problems. See the web site.
References on Fourier's model, eigenanalysis.
What's eigenanalysis slides 2008 ( pdf)
algebraic eigenanalysis slides 2008 ( pdf)
Eigenanalysis-2 slides f2008 (pdf)
Eigenanalysis-I manuscript S2007 (typeset 19 pages, 200k pdf)
25 Nov:
Lecture: 7.1, 7.2, 7.3. Systems u'=Au with 2x2 constant matrix A.
Scalar methods based on the characteristic equation det(A - lambda I)=0.
Shortcut for a 2x2 system u'=Au with A not a diagonal matrix. Matrix
eigenanalysis method. Proof for the 2x2 case. How to convert
mx''+cx'+kx=F0 cos (omega t) into a dynamical system u'=Au+F(t).
Electrical systems u'=Au+E(t) from LRC circuit equations.
Solving for eigenpairs, 2x2 matrix A. Fourier's model, diagonalization,
eigneanalysis method of ch7.
26 Nov:
Lecture: 5.6. Equations mx''+kx=F0 cos(omega t) for pure resonance
and beats, equation mx''+cx'+kx=F0 cos(omega t) for practical resonance.
Formulas for the unique periodic solution
x=A cos(omega t) + B sin(omega t)
and its amplitude
C = F0/sqrt(delta), where delta=(k-m omega)^2+(c omega)^2
Pure and practical resonance formula
omega = sqrt(k/m - c^2/(2m^2))
THEOREM. The homgeneous solution has limit zero at infinity for c>0.
THEOREM. Three is a unique periodic solution obtained by undetermined
coefficients when c>0 and the period is 2 PI / omega. This solution is
the observable or steady state. All solutions of the homogeneous
equation are transient.
In the Laplace solution, the output L(x) equals the transfer function
time the Laplace of the input F0 cos(omega t). It is possible to find the
steady state solution by dropping all negative exponential terms in the
Laplace solution.
Graphics for beats [x=sin(10 t)sin(t/2)], slowly-oscillating envelope,
rapidly oscillating harmonic with time-varying amplitude. Pure resonance
and explosion of the solution, versus practical resonance, where omega
is tuned to make C(omega)=maximum among all input frequencies.
Resonance examples: Soldiers marching in cadence, Tacoma narrows bridge,
Wine Glass Experiment. Von Karmon and vortex shedding. Cable model of the
year 2000. Resonance explanations.
Plot of C(omega) versus omega. Practical resonance as the high point
on a graphic. Maple lab on the Tacoma narrows (extra credit).
Week 12, Nov 17 to 21: Sections 10.4,10.5,6.1,6.2
17 Nov:
Lecture: Topics from Laplace theory.
Periodic function theorem. Convolution theorem. Convolution examples. How to do 10.4 problems.
Projection slides:
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
Laplace Table Derivations.
(pdf slides)
Laplace rules. Includes: Lerch, Linearity,
t-Differentiation, s-Differentiation, Integral, Shifting I and II, Periodic, Convolution.
(pdf slides)
Electric circuits. Transfer function. Impedance Z. Reactance S=omega L - (1/C)(1/omega)
Time lag delta/omega. Steady-state periodic solution. Transient solution.
Review: Electrical-mechanical analogy. Faraday's law, Ohm's law, Coulomb's law,
Kirchhoff's laws. Voltage drop formulas. Number of independent differential equations used to describe
a multi-loop circuit.
Electrical resonance.
Reference: Edwards-Penney, Differential Equations and Boundary Value
Problems, 4th edition, section 3.7. Extra pages supplied by Pearson with
bookstore copies of the 2250 textbook. Also available as a xerox copy in
case your book came from elsewhere [send email].
Review: dampener design in a rotating machine with flywheel.
References and slides:
Electrical circuits.
(pdf slides)
Laplace rules. Includes: Lerch, Linearity,
t-Differentiation, s-Differentiation, Integral, Shifting I and II, Periodic, Convolution.
(pdf slides)
Laplace table derivations.
(pdf slides)
18 Nov:
Drill: Solving second order DE by Laplace. What to expect and how to do it.
Example: y'' = t, y(0)=0, y'(0)=1.
Impedance, reactance, electrical resonance. Steady-state current amplitude.
Transfer function. Input and output equation.
Solving a 2x2 dynamical system using Laplace's method. Resolvent method example.
Methods for solving a 2x2 dynamical system: Ch1 method /w det(A-lambda
I)=0, Laplace resolvent, eigenanalysis, Putzer's method for the matrix
exponential.
Laplace method for systems, resolvent.
(pdf slides)
Lecture: Topics from Laplace theory, linear systems and electric circuits.
Brine tank models. Recirculating brine tanks. Pond pollution.
All are 3x3 system applications that can be solved by Laplace methods.
Brine tank cascade. Brine tank recyling.
(pdf slides)
Lecture: Systems of two differential equations, Cramer's Rule,
matrix inversion methods. The Laplace resolvent method for systems.
19 Nov:
Drill problems: More on ramp, sawtooth, staircase, rectified sine.
Problem: Write a 2x2 dynamical system as a vector-matrix equation u'=Au.
Problem: Solve a 2x2 dynamical system in vector-matrix form u'=Au.
Review: Expansions of periodic functions as s-domain Laplace transforms.
The periodic function theorem.
Laplace of the square wave, tanh function.
Home heating example.
Home heating: attic, main floor, basement.
(pdf slides)
Lecture: Intro to eigenanalysis, ch6. Examples and motivation.
Fourier's model. History.
References on Fourier's model, eigenanalysis.
What's eigenanalysis slides 2008 (pdf)
Eigenanalysis-I manuscript S2007 (typeset 19 pages, 200k pdf)
21 Nov:
Review and Drill: Laplace of the unit step, pulse and ramp.
Derivations from the integral theorem.
Laplace theory tricks with the three Shifting theorems using
piecewise continuous functions. Solutions to 10.5-4,22,28 in class
and also problem notes added to the web site.
Lecture: Intro to eigenanalysis, ch6. Data conversion example.
Fourier's model. History from 1822.
Slides projected today:
Piecewise-defined functions and Laplace theory slides F2008 (pdf)
What's eigenanalysis slides 2008 (pdf)
Week 11, Nov 10 to 14: Sections 10.1,10.2,10.3,10.4
10 Nov:
Lecture by Damon Toth: Ch10, sections 10.2,10.3
Lecture: Basic Laplace theory. Shift theorem. Parts theorem.
Forward table. Backward table. Extensions of the Table.
Lecture: Solving differential equations by Laplace's method.
Rules and the brief table [Laplace calculus]. Partial fractions.
Using trig identities [sin 2u = 2 sin u cos u, etc].
11 Nov:
Lectures resume by Grant Gustafson. Slides on Laplace theory. Worked examples
and problems from 10.1, 10.2.
Intro to Laplace theory. L-notation. Forward and backward table. Examples.
(pdf slides)
Reference: Heaviside's method
2008, typeset
12 Nov:
Lecture: Partial fraction expansions suited for LaPlace theory.
Solving initial value problems by LaPlace's method. Details of
the backward table and the forward table. Information about the
equivalence of the inverse of L and Lerch's theorem.
Use of inverse Laplace and Lerch's theorem.
History of Laplace calculus and Newton Calculus.
Newton integral calculus and Laplace calculus. Laplace method.
(pdf slides)
Laplace theory references
Intro to Laplace theory. L-notation. Forward and backward table. Examples.
(pdf slides)
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
Laplace rules. Includes: Lerch, Linearity,
t-Differentiation, s-Differentiation, Integral, Shifting I and II, Periodic, Convolution.
(pdf slides)
Reference: Laplace theory typeset manuscript 2008
(49 pages pdf)
Reference: Heaviside's method
2008, typeset
(pdf)
Ch10 Laplace exercise solutions [scanned], 10.1 to 10.4
(9 pages, 2mb, pdf)
Projection Slides:
Newton integral calculus and Laplace calculus. Laplace method.
(pdf slides)
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
Laplace rules. Includes: Lerch, Linearity,
t-Differentiation, s-Differentiation, Integral, Shifting I and II, Periodic, Convolution.
(pdf slides)
Laplace method for systems, resolvent.
(pdf slides)
Brine tank cascade. Brine tank recyling, home heating.
(pdf slides)
Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation.
(pdf slides)
13 Nov:
Gully and Shiu: Problem session and drill on Laplace theory.
14 Nov:
Lecture: Review and drill on the forward and backward table, Laplace rules.
Examples of forward and backward table calculations. Harmonic oscillator.
Rotating machine with one piston and massive flywheel - damping design.
Design of physical models from Laplace models instead of differential equations.
The transfer function. Input and output in the s-domain.
Periodic function theorem. Convolution theorm. Variation
of parameters revisited: the algebraic solution of a DE in integral form
from the convolution theorem:
x''+x=F(t),x(0)=x'(0)=0 ==> x(t) = int(sin(t-u)F(u)du,u=0..t).
Details of proof for the convolution theorem.
Periodic waves used in engineering. Step, Ramp, Rectified sine wave,
sawtooth, staircase.
Partial fractions, Heaviside
method, shortcuts, failsafe method for partial fractions [==sampling method],
the method of atoms, Heaviside coverup.
Week 10, Nov 3 to 7: Sections 5.3,5.4,5.5,10.1
03 Nov:
Problems solved in class: all of 5.2, 5.3 and 5.4-20,34
Lecture 5.4: overdamped, critically damped and under damped behavior, pseudoperiod.
Variation of parameters. Solved examples. Problems 5.5-54,58.
Start undetermined coefficients.
04 Nov:
Solutions to 5.2, 5.3, 5.4, 5.5 problems.
Undetermined coefficients. Finding a characterisitic polynomial p(r)
for a DE such that a given linear combination of atoms f(x) is a solution.
How to find the homogeneous solution yh from a characteristic equation.
Given a trial solution with undetermined coefficients, find a system of
equations for d1, d2, ... and solve it.
Euler's theorem forward and backward.
How to compute the trial solution in undetermined coefficients.
05 Nov:
Exam review: questions answered about exam 2.
Sample test for problems 1,4,5.
Lecture and review: Standard basis in R^n. Theorems on independent sets and bases.
Kernel Thm. Subspace proofs and the kernel theorem. Basis for the solution space of Ax=0.
Review independence.
Review of Pivot, RREF [RANK] and DETERMINANT test for
independence of fixed vectors.
Orthogonality. General vector spaces.
Lecture: Picard's Theorem and the dimension of the solution space
of a linear constant system of differential equations. How to solve
y''' + y' = x - x^2 by undetermined coefficients.
06 Nov:
11 Mar: Exam 2 at 7:00am in WEB 103.
07 Nov:
Lecture: Adam Gully, 10.1.
Reading: Continue into chapter 10. We return to 5.6, ch6, ch7, ch8, ch9 later.
Lecture: Introduction to Laplace's method. The method of quadrature
for higher order equations and systems. Calculus for chapter one
quadrature versus the Laplace calculus. The Laplace integrator
dx=exp(-st)dt. The Laplace integral abbreviation L(f(t)) for
the Laplace integral of f(t). Lerch's cancellation law and the fundamental
theorem of calculus.
Def: Direct Laplace transform == Laplace integral ==
int(f(t)exp(-st),t=0..infinity) == L(f(t)).
Linearity. The s-differentiation theorem (d/ds)L(f(t))=L((-t)f(t)).
Laplace theory references week 10-11.
Intro to Laplace theory. L-notation. Forward and backward table. Examples.
(pdf slides)
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
Laplace theory typeset manuscript 2008
(49 pages pdf)
Heaviside's method
2008, typeset (pdf)
Ch10 Laplace solutions [scanned], 10.1 to 10.4
(9 pages, 2mb, pdf)
Week 9, Oct 27 to 31: Sections 5.3,5.4,5.5,10.1
27 Oct:
Linear Algebra Review.
Theorem: pivot columns are independent and non-pivot columns
are linear combinations of the pivot columns.
Theorem: rank(A)=rank(A^T).
Theorem: A set of nonzero pairwise orthogonal vectors is linearly independent.
In-class solutions. Please read the problem notes, ch4.
Problem session 4.5, 4.6, 4.7.
Web References:
Lecture slides on Vector spaces, Independence tests. (pdf)
Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)
Lecture slides on orthogonality, independence of orthogonal sets, Cauchy-Schwartz, Pythagorean Identity (pdf)
Linear DE Slides.
How to solve linear DE, examples of orders 1 to 4, Euler's theorem explained. slides 2008 (pdf)
How to solve examples like y''=0, y''+3y'+2y=0, y''+y'=0, y'''+y'=0.
28 Oct:
Linear DE
Higher order constant equations, homogeneous and
non-homogeneous structure. Superposition. Picard's Theorem.
Solution space structure. Dimension of the solution set.
Euler's theorem. What to do with complex roots and conjugate pairs of
factors (r-a-ib), (r-a+ib).
The formula exp(i theta)=cos(theta) + i sin(theta). How to solve homogeneous
equations: use Euler's theorem to find a list of n distinct atoms that are solutions
to the equation. Specific examples for first, second and higher order
equations.
How to use Euler's theorem to construct atoms of a linear
differential equation.
Common errors in solving higher order equations.
Solved examples like the 5.1,5.2,5.3 problems.
Identifying atoms in linear combinations.
29 Oct:
Due today: Page 294, 5.1: 34, 36, 38, 40, 42, 46, 48
Problem session on 5.1, 5.2, 5.3, 5.4.
How to solve for c1, c2, etc when initial conditions are given.
Theory of equations and 5.3-32.
Solving a DE when the characteristic equation has complex roots.
Higher order equations or order 3 and 4.
Damped and undamped equations. Phase-amplitude form.
Partly solved 5.4-34. The DE is 3.125 x'' + cx' + kx=0. The
characteristic equation is 3.125r^2 + cr + kr=0 which factors into
3.125(r-a-ib)(r-a+ib)=0 having complex roots a+ib, a-ib. Problems 32, 33
find the numbers a, b from the given information. This is an inverse
problem, one in which experimental data is used to discover the
differential equation model. The book uses its own notation for the
symbols a,b: a--> -p and b --> omega. Because the two roots a+ib, a-ib
determine the quadratic equation, then c and k are known in terms of
symbols a,b.
Finding 2 atoms from one complex root. Why the complex conjugate root
identifies the same two atoms.
Equations with both real roots and complex roots.
An equation with 4 complex roots. How to find the 4 atoms.
Review and Drill.
The RLC circuit equation and its physical parameters.
Spring-mass equation mx''+cx'+kx=0 and its physical parameters.
Solving more complicated homogeneous equations. Example: Linear DE given by
roots of the characteristic equation. Example: Linear DE given by factors of
the characteristic polynomial. Example: Construct a linear DE of order 2 from
a list of two atoms that must be solutions. Example: Construct a linear DE from roots of
the characteristic equation. Example: Construct a linear DE from its general solution.
30 Oct: Gully and Shiu reviewed the last 3 problems on the
sample midterm for next week:
Spring 2008 7:30 exam 2 solution key (pdf)
31 Oct:
Due today: 5.2-18,22
Problem session. Includes some 5.5 problems.
Lecture: 5.4 Damped and undamped motion. Pendulum, harmonic oscillations,
spring-mass equation, phase-amplitude conversions from the trig course.
Beats. Pure resonance. Cafe door. Pet door. Car on 4 springs.
Variation of parameters formula (33) in 5.5. Undetermined coefficients.
References: 5.5 variation of parameters formula (33).
Second order variation of parameters slides 2008 (pdf)
Second order variation of parameters (typeset, 6 pages pdf)
References: Sections 5.4, 5.6. Forced oscillations.
Forced vibrations, undamped case, slides 2008 (pdf)
Forced vibrations, damped case, slides 2008 (pdf)
Forced vibrations and resonance, slides 2008 (pdf)
Start section 5.5 undetermined coefficients today.
References for weeks 9-10.
Picard's Theorem for systems, slides 2008 (pdf)
How to solve linear DE, examples of orders 1 to 4, Euler's theorem explained. slides F2008 (pdf)
How to solve linear DE, slides S2008 (pdf)
Solving linear DE, examples of orders 1 to 4, slides 2008 (pdf)
Basic Reference: First order constant coefficient recipe, structure of solutions, superposition (slides, 3 pages pdf)
Basic Reference: First order constant coefficient recipe + theory + variation of parameters and undetermined coefficients (typeset, 11 pages pdf)
Second order constant coefficient recipe + theory (typeset, 7 pages pdf)
Undetermined coefficients example, cafe door, pet door, phase-amplitude, resonance slides F2007
Week 9 references for Edwards-Penney section 5.5
Second order variation of parameters (typeset, 6 pages pdf)
Second order undetermined coefficients (typeset, 7 pages pdf)
Higher order linear differential equations. Higher order recipe. Higher order undetermined coefficients. (typeset, 9 pages pdf)
Undetermined coefficients slides Nov 9, 2008(pdf)
Ch5 undetermined coeff illustration (3 pages, pdf)
Week 8, Oct 20 to 24: Sections 4.4,4.5,4.6,4.7,5.1,5.2,5.3
20 Oct:
Lecture: More on independence. Algebraic tests. Geometric tests.
General solutions with a minimal
number of terms.
Data recorder example and data conversion to fit physical models.
Basis == independence + span. Independence of atoms.
Function spaces. General solution and shortest answer.
The pivot theorem. Rank test. Determinant test.
Sampling test. Wronskian test.
21 Oct: Collected 4.1
Subspaces. Using the kernel theorem. Problems 4.1,4.2.
Lecture: Constant coefficient equations, spring-mass system, harmonic oscillation,
damped and undamped systems, forced systems. RLC circuit equation. Examples of
order 2,3,4.
Spring-mass DE and RLC-circuit DE derivations.
Electrical-mechanical analogy.
Lecture: Definition of atom. Independence of atoms.
Applications to be done later: Method of atoms in partial fractions.
Sampling in partial fractions. Heaviside's coverup method.
22 Oct: Collected 4.2
Picard's Theorem for higher order DE and systems.
Solution space theorem for linear differential equations.
For constant equations, the solution is a linear combination of atoms.
Dimension of the solution space. Structure of solutions.
Announcement: Euler's theorem and how to solve any homogeneous linear constant DE.
Week 8 references.
Picard's Theorem for systems, slides 2008 (pdf)
How to solve linear DE, examples of orders 1 to 4, Euler's theorem explained. slides 2008 (pdf)
Reference: How to solve linear DE, slides 2008 (pdf)
Reference: Solving linear DE, examples of orders 1 to 4, slides 2008 (pdf)
Basic Reference: First order constant coefficient recipe, structure of solutions, superposition (slides, 3 pages pdf)
Basic Reference: First order constant coefficient recipe + theory + variation of parameters and undetermined coefficients (typeset, 11 pages pdf)
Ch5. Constant coefficient recipe (typeset, 2 pages, pdf)
Second order constant coefficient recipe + theory (typeset, 7 pages pdf)
Undetermined coefficients, cafe door, pet door, phase-amplitude, resonance slides F2007
24 Oct: Collected 4.3
Linear Algebra review: Solutions to 4.3-18,24, 4.4-6,24 and
4.7-10,22,26.
Dimension. Basis for linear
system Ax=0 from the last frame algorithm. Partial derivatives and bases.
Proofs involving subspaces for vector spaces V whose
data item packages are functions or abstract vectors.
Lecture: Second order and higher order differential Equations.
Picard theorem for second order equations, superposition, solution
space structure, dimension of the solution set.
Euler's theorem. Quadratic equations
again. Constant-coefficient second order homogeneous differential
equations. Characteristic equation and its factors determine the atoms
for a second order linear DE.
Sample equations: y''=0, y''+2y'+y=0, y''-4y'+4y=0, y''+y=0, y''+3y'+2y=0,
mx''+cx'+kx=0, LQ''+RQ'+Q/C=0.
Week 7, Oct 6 to 10: Sections 3.6,4.1,4.2,4.3
06 Oct:
Review: MCI-SPRINT-ATT power method example. History of telecoms 1984 to present.
Brief telecom history. (txt)
Lawrence Page's pagerank algorithm, google web page rankings, relation
to the power method, stochastic matrices and eigenanalysis. Reference:
Where to find articles about Page's algorithm. (txt)
Lecture: Cofactor expansion. Hybrid methods. Frame sequences and
determinants. Formula for det(A) in terms of swap and mult operations.
Hybrid methods to compute a determinant. In-class example.
Special theorems for determinants having a zero row, duplicates rows or
proportional rows.
How to use the 4 rules to compute det(A) for any size matrix.
Computing determinants of sizes 3x3, 4x4, 5x5 and higher.
Elementary matrices and determinants. Determinant product rule.
Cramer's rule. Adjugate matrix. In-class examples on Cramer's rule for
2x2 and 3x3. How to form minors, cofactors and the adjugate matrix.
Statements of the determinant product theorem and the adjugate inverse theorem.
07 Oct:
Lecture: Cofactor rule and the adjugate matrix. How to find det(A) from A and
adj(A). Cofactor rules imbedded in the formula det(A)I = A adj(A).
Determinant slides 2008 (pdf)
Examples in class: (1) solve a 2x2 system by Cramers rule. (2) Find entry in row 3, col 2
of the inverse of A = adj(A)/det(A) as a quotient of 2 determinants. (3)
Find det(A) from A and adj(A).
Problem session on 3.6 exercises.
08 Oct:
Review: 2x2, 3x3, 4x4, 5x5 determinant evaluation examples.
Cramers rule example. Adjugate example. Computing entry 2,3 in adj(A) or
inverse(A). How to reduce the Four rules [triang, swap , combo, mult] to
Two Rules using the determinant product theorem det(AB)=det(A)det(B).
Lecture: Chapter 4, sections 4.1 and 4.2.
Web references for chapter 4.
Lecture slides on Vector spaces, Independence tests. (pdf)
Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)
Vector space, Independence, Basis, Dimension, Rank (typeset pdf)
Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9 (4 pages, 480k pdf)
Vectors==packages of data items.
Examples of vectors: digital photos, Fourier coefficients, Taylor
coefficients, sols to DE like y=2exp(-x^2). How to develop notation for
functions that parallels column vectors. Hybrid vector spaces and RLC circuits.
The toolkit of 8 properties (Thm 1, 4.2). Vector spaces.
09 Oct:
Thursday Lecture [Gully and Shiu]:
Maple Lab 3, Numerical Solutions
Maple Lab 3 Spring 2007 (pdf)
Maple L3 snips Spring 2007 (maple text)
Maple Worksheet files: In Mozilla firefox, save to disk using
right-mouseclick and then "Save link as...".
Some browsers require SHIFT and then mouse-click. Open the saved
file in xmaple.
Maple L3 snips worksheet Spring 2007 (maple .mws)
Numerical DE coding hints (txt)
The actual symbolic solution derivation and answer check are
submitted as L3.1. Confused? Follow the details in the next link.
Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)
Do not submit any work for L3.1 with decimals! Only 1.4 methods are
to appear.
The numerical work, Euler, Heun, RK4 parts are L3.2, L3.3, L3.4.
Confused about what to put in your L3.2 report? Do the same
as what appears in the sample report for 2.4-3 (below).
Sample Report for 2.4-3. Includes symbolic solution report. (pdf 3 pages, 350k)
Sample maple code for Euler, Heun, RK4 (maple .mws)
Sample maple code for exact/error reporting (maple .mws)
Additional reference, probably not needed:
Report details on 2.4,2.5,2.6 prob 6 (pdf)
10 Oct:
Didn't do 3.6-64? Do the extra credit from ch3.
Abstract vector spaces. Working set == subspace. Vector space == data set.
Lecture: Chapter 4, section 4.3
More on vector spaces and subspaces: detection of subspaces and data
sets that are not subspaces.
Theorems: Subspace criterion, Kernel theorem, Not a subspace theorem.
Use of theorems 1,2 in section 4.2. Solutions of problems 4.1, 4.2.
Example in class: Avoid using the subspace criterion on the set S in
R^3 defined by x+y+z=0, by writing it as Ax=0, followed by applying the
kernel theorem (thm 2 page 239 of Edwards-Penney).
More on the toolkit. Vectors as packages of data items. Examples
of vector packaging in applications. The kernel: sols of Ax=0.
Lecture: Independence and dependence. Atoms for differential equations.
Geomtric test for one or two vector independence.
Web References:
Lecture slides on Vector spaces, Independence tests. (pdf)
Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)
Week 6, Sept 29 to Oct 3: Sections 3.4,3.6,3.6,4.1
29 Sep:
Digitial photos and matrices.
Web Reference: Image sensors, digital photos, checkerboard analogy,
visualization of matrix addition and scalar multiplication.
Digital photos and matrix operations slides S2008
Digital photo slides F2008
The four vector models, vector spaces and the 8-property toolkit.
Properties of matrices. Matrix multiply rules. Linear systems as the matrix equation Ax=b.
30 Sep:
Lecture: Elementary matrices.
Theorem: rref(A)=(product of elementary matrices)A.
Web Reference: Elementary matrices
Elementary matrix slides S2008
Lecture: How to write a frame sequence as a product of elementary matrices.
Lecture: How to compute the inverse matrix from inverse = adjugate/determinant
and also by frame sequences. Inverse rules.
Web Reference: Construction of inverses. Theorems on inverses.
slides on rref inverse method S2008
01 Oct:
[on Monday, all of 3.4 problems will be due, 3.4-20,30,34,40]
Lecture:
Discussion of Cayley-Hamilton theorem [3.4-29] and how to solve problem 3.4-30.
Discussion of 3.4 problems.
Due next Week: The rest of 3.4; see FAQ 3.4 for details.
Problems 3.4-17 to 3.4-22 are homogeneous systems Ax=0 with A in
reduced echelon form. Apply the last frame agorithm then write the
general solution in vector form.
Problem 3.4-29 is used in Problem 3.4-30. The result is the Cayley-Hamilton Theorem,
a famous theorem of linear algebra which is the basis for solving systems of differential equations.
Problem 3.4-40 is the superposition principle for the matrix equation Ax=b. It is the analog of the differential equation
relation y=y_h + y_p.
03 Oct:
Please work on 3.6 and 4.1 problems.
Inverses of elementary matrices. Solving B=E3 E2 E1 A for matrix A.
About problem 3.5-44: This problem is the basis for the
fundamental theorem on elementary matrices (see below). While 3.5-44 is
a difficult technical proof, the extra credit problems on this subject
replace the proofs by a calculation. See Xc3.5-44a and Xc3.5-44b.
Lecture: Introduction to 3.6 determinant theory and Cramer's rule.
Lecture: Adjugate formula for the inverse. Review of Sarrus' Rules.
slides for 3.6 determinant theory
Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)
Determinants and Cramer's Rule (typeset, 11 pages, 140k pdf)
Determinant slides 2008 (pdf)
Methods for computing a determinant. Sarrus' rule,
four rules for determinants, cofactor expansion, hybrid methods.
In-class examples.
Triangular rule [one-arrow Sarrus' rule], combo, swap and mult rules.
College algebra determinant definition and Sarrus' rule for 2x2 and 3x3 matrices.
Examples where det(A) can be comupted.
Determinant values for elementary matrices: det(E)=1 for combo(s,t,c),
det(E)=m for mult(t,m), det(E)=-1 for swap(s,t).
Main theorems:
- Computation by the 4 rules, cofactor expansion, hybrid methods.
- Determinant product theorem det(AB)=det(A)det(B).
- Cramer's Rule for solving Ax=b.
- Adjugate inverse formula inverse(A) = adjugate(A)/det(A).
Week 5, Sept 22 to 26: Sections 3.1,3.2,3.3,3.4
22 Sep:
Lecture: 3.1, frame sequences, combo, swap, multiply, geometry
Prepare 3.1 problems for next collection.
References for chapter 3
Slides on Linear equations, reduced echelon, three rules (pdf)
Slides on Linear equations, unique solution case (pdf)
Slides on Linear equations, no solution case, signal equations (pdf)
Slides on Linear equations, infinitely many solution case, last frame algorithm (pdf)
Three rules, frame sequence, maple syntax (typeset, 7 pages, pdf, 12 Oct 2006)
Frame sequence and general solution, 3x3 system (1 page, pdf, 28-Sep-2006)
Linear algebra, no matrices, DRAFT 8Feb2008 (typeset, 44 pages, pdf)
Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)
vectors and matrices (typeset, 14 pages)
Matrix equations (typeset, 12 pages)
Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)
Elementary matrices, vector spaces, slides (8 pages, pdf, 12 Oct 2006)
Determinants and Cramer's Rule (typeset, 11 pages, 140k pdf)
23 Sep: Due today, symbolic sol L3.1, L4.1.
Lecture: 3.1, 3.2, 3.3, frame sequences, general solution, three possibilities.
A detailed account of the three possibilities. How to solve a linear system
using the tookit [swap, combo, mult] and frame sequences, for the unique solution case,
no solution case and infinitely many solution case. Examples.
Slides for this lecture.
Slides on Linear equations, unique solution case (pdf)
Slides on Linear equations, no solution case, signal equations (pdf)
Slides on Linear equations, infinitely many solution case, last frame algorithm (pdf)
Sample solution L3.1 (jpg)
In 3,2 solutions, back-substitution should be presented as combo
operations in a frame sequence, not as isolated algebraic jibberish.
Computer algebra systems and error-free frame sequences.
How to program maple to make a frame sequence without errors.
Problem 3.2-24: See
Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)
Beamer Slides on three possibilities with symbol k, Sept2007 (9 pages,pdf)
See also Example 10 in
Linear equations, no matrices, DRAFT Feb2008 (typeset, 44 pages, pdf)
Prepare 3.3 problems 8, 18 for next time. Please use frame sequences
to display the solution, as in today's lecture examples. It will be a
sequence of augmented matrices. Yes, you may use maple to make the frame
sequence and to do the answer check [rref(A);].
24 Sep:
Due today, Page 152, 3.1: 4, 18, 26
Answer checks should also use the online FAQ.
Lecture: 3.3 and 3.4.
Translation of equation models to (augmented) matrix models and back.
Combo, swap and multiply for matrix models. Frame sequences for matrix
models.
Review of rref or last frame test.
Frame sequences and the Vector form of the
solution to a linear system. Matrix multiply and the equation Ax=b.
Slides on matrix Operations (pdf)
Equation ideas can be used on a matrix A. View matrix A
as the set of coefficients of a homogeneous linear system Ax=0. The augmented
matrix B for this homogeneous system would be the given matrix with a
column of zeros appended: B=aug(A,0).
General structure of linear systems. Superposition. Last frame algorithm.
Review and drill of last frame algorithm and the scalar and vector forms of the solution.
25 Sep:
Shiu reviews Sample Midterm problems 1-5. More details on maple lab 2.
26 Sep: Due in class, Page 162, 3.2: 10, 18, 24
Maple lab problm L2.1 discussed today. Solution projected for L2.1 and L2.4 solutions.
Happens only in class, no web solutions available.
Continue lecture on 3.2, 3.3. Matrix notation introduced.
Ideas of rank, nullity, dimension in examples.
Lecture: The 8-property toolkit for vectors. Vector spaces. Read 4.1
in Edwards-Penney, especially the 8 properties pages 223-226 [227-233 can be read later].
Review of vector models is in the slide set
Slides on vector models and vector spaces 2008 (pdf)
Slides and examples for chapter 3
Linear equations, reduced echelon, three rules (typeset, 7 pages, pdf)
Three rules, frame sequence, maple syntax (typeset, 7 pages, pdf, 12 Oct 2006)
Snapshot sequence and general solution, 3x3 system (1 handwritten page, pdf, 28-Sep-2006)
Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)
Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)
Elementary matrices, vector spaces, slides (8 pages, pdf, 12 Oct 2006)
Slides on matrix Operations (pdf)
Typeset references for ch3 and ch4
Linear equations, no matrices, DRAFT 2008 (typeset, 44 pages, pdf)
vectors and matrices (typeset, 14 pages)
Matrix equations (typeset, 12 pages)
Determinants and Cramer's Rule (typeset, 11 pages, 140k pdf)
Review of linear equations: Rank, Nullity, dimension, 3 possibilities, elimination algorithm.
Slides on rank, nullity, elimination algorithm 11FEb2008 (pdf)
Slides on the 3 possibilities, rank, sytems with symbol k 11Feb2008 (pdf)
Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)
Lecture: 3.2, 3.3. Intro to matrices and matrix models for linear equations. Matrix multiply Ax for x a vector.
Fixed vectors, physics vectors i,j,k, engineering vectors (arrows), Gibbs vectors.
Parallelogram law. Head-tail rule.
Answer to the question: What did I just do, when I found rref(A)?
Week 4, Sep 15-19: Sections 2.3,2.4,2.5,2.6, 3.1, 3.2
15 Sep: Due today 2.2-10,14. All of 2.3 due Wed, followed by
Maple Lab 1: Intro maple L1.1, L1.2.
If you are unable to turn in this lab, then see the Ch2 Extra Credit problems,
which contains 2 problems like L1.1 and L1.2.
Review and reading: Please study the slides on partial fractions.
Partial Fraction Theory 2008(125k pdf)
Nonlinear air resistance models F=kx'|x'|.
Lecture slides on the reading assignment for 2.3, and the work of Isaac
Newton on ascent and descent models for kinematics with air resistance.
Newton models, projectile slides 2008 (pdf)
Problem notes for 2.3-10. 2.3-20 are available,
Click Here.
16 Sep: Exam 1 review, questions and examples on problems 1,2,3,4,5.
For more details on the 2.3 problems,
Click Here.
Intro to the Jules Verne problem and its solution.
Slides on the Jules Verne problem [reading: 2.3].
Earth to the moon slides 2008 (pdf)
Problems discussed: 2.3-10 and 2.3-20.
Problem notes for 2.3-10. 2.3-20 including sample maple code:
Chapter 2, 2.3-10,20,22 notes S2007
Reading assignment: proofs of 2.3 theorems in the textbook and derivation of
details for the rise and fall equations with air resistance.
17 Sep: Exam 1 review, questions and examples on problems 1,2,3,4,5.
Due today, 2.3: 10, 20.
Lectures begin for 2.4, 2.5, 2.6 topics on numerical solutions.
Numerical DE slides 2008 (14 slides pdf)
Introduction to
numerical solutions of quadrature problems y'=F(x), y(x0)=y0.
The examples used in maple labs 3, 4 are
y'=-2xy, y(0)=2, y=2exp(-x^2) and y'=(1/2)(y-1)^2, y(0)=2,
y=(x-4)/(x-2). Web notes (item 2 in the references below) contain the
examples y'=3x^2-1, y(0)=2, y=x^3-x+2 and y'=exp(x^2), y(0)=2 with
solution y=int(F,0..x)+y0, F(x)=exp(x^2).
Intro to Rect, Trap, Simp rules
from calculus and Euler, Heun, RK4 rules
from this course.
References for numerical methods:
Numerical DE slides 2008 (14 slides pdf)
Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
Sample Report for 2.4-3 (pdf 3 pages, 350k)
ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)
The work for 2.4, 2.5, 2.6 is in maple lab 3 and maple lab 4. Details for lab 3:
Maple Lab 3, Numerical Solutions
Maple Lab 3 F2008 (pdf)
Maple L3 snips F2008 (maple text)
Maple Worksheet [.mws] files: In Mozilla firefox, save to disk using
right-mouseclick and then "Save link as...".
Some browsers require SHIFT and then mouse-click. Open the saved
file in xmaple or maple.
Maple L3 snips worksheet Spring 2008 (maple .mws)
Numerical DE coding hints (txt)
The actual symbolic solution derivation and answer check are
submitted as L3.1. Confused? Follow the details in the next link.
Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)
Do not submit any work for L3.1 with decimals! Only 1.4 methods are
to appear.
The numerical work using Euler, Heun, RK4 appears in L3.2, L3.3, L3.4.
Confused about what to put in your L3.2 report? Do the same
as what appears in the sample report for 2.4-3 (below).
Sample Report for 2.4-3. Includes symbolic solution report. (pdf 3 pages, 350k)
Download all .mws maple worksheets to disk, then run in maple.
Sample maple code for Euler, Heun, RK4 (maple .mws)
Sample maple code for exact/error reporting (maple .mws)
18 Sep: Gully and Shiu, Exam 1 review, questions and examples on problems 1,2,3,4,5.
Discuss questions on maple Lab 2, especially parts 2,3,4.
How to present the solutions to the math problems L3.1, L4.1 [no decimals,
computers or numerical work!].
19 Sep: Nothing due today, catch-up day, switching chapters.
Continue the lecture on numerical methods.
Discussed y'=3x^2-1, y(0)=2 with solution y=x^3-x+2. Dot tables, connect the dots graphic.
How to draw a graphic without knowing the solution equation for y.
Main example y'=srqt(x)exp(x^2), y(0)=2. Making the dot table by approximation
of the integral of F(x). Rect, Trap, Simp rules and their accuracy of 1,2,4 digits resp.
Example for your study: The problem y'=x+1, y(0)=1 has a dot table with
x=0, 0.25, 0.5, 0.75, 1 and y= 1, 1.25, 1.5625, 1.9375, 2.375. The exact
solution y=1/2+(x+1)^2/2 has values y=1, 1.28125, 1.625, 2.03125,
2.5000. Try to determine how the dot table was constructed and identify which rule [Rect, Trap, Simp] was applied.
Symbolic solution, no numerics, maple L3.1, L4.1 due next Monday.
Discussion of Euler, Heun, RK4 algorithms. Computer implementations.
Numerical work maple L3.2-L3.4, L4.2-L4.4 will be submitted after the spring break.
All discussion of maple programs will be based in the Tuesday session.
There will be one additional presentation of maple lab details in the main lecture.
References for numerical methods:
Numerical DE slides 2008 (14 slides pdf)
Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
Sample Report for 2.4-3 (pdf 3 pages, 350k)
ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)
Maple lab symbolic sol L3.1, L4.1 will be due Monday. Exam 1 on Oct 2.
Week 3, Sept 8 to 12: Sections 1.5, 2.1, 2.2, 2.3.
08 Sep:
Collected Page 41, 1.4: 18, 22, 26
Linear integrating factor method 1.5. Application to y'+2y=1. Testing
linear DE y'=f(x,y) by f_y independent of y. Examples of linear equations
and non-linear equations. Integrating factor Lemma. Main theorem on linear DE
and explicit general solution.
Three linear examples: y'+(1/x)y=1, y'+y=x, y'+2y=1.
Pharmokinetics for drug transport, brine tanks, pollution models.
References for linear DE:
Linear integrating factor method, Section 1.5, slides (pdf)
Linear DE method, Section 1.5 slides: 1.5-3,5,11,33+Brine mixing (9 pages, pdf)
Linear DE part I (Integrating Factor Method), (typeset, 8 pages, pdf)
Linear DE part II (Variation of Parameters, Undetermined Coefficients), (typeset, 7 pages, pdf)
How to do a maple answer check for y'=y+2x (TEXT 1k)
Linear first order slides, integrating factor method (2 pages, pdf)
09 Sep:
Nothing collected in class.
Section 1.5. General solution of the homogeneous equation. Superposition
principle. See FAQ and slides for exercises 1.5-3, 1.5-5, 1.5-11, 1.5-34.
Some more class discussion of 1.5-34.
On 1.5-34: The units should be taken as millions of cubic feet.
The textbook gives the initial value problem x'=r_i c_i - (r_0/v)x,
x(0)=x_0. The initial value is x_0 =
(0.25/100)8000, the output rate is r_0=500, and the tank volume is V=8000.
Please determine the value for the
input concentration, constant c_i.
You should obtain r_ic_i=1/4.
Then solve the initial value problem.
The book's answer t = 16 ln 4 = 22.2 days is correct.
Due Wed, Page 54, 1.5: 8, 18, 20, 34.
Introduction to 2.1, 2.2 topics: autonomous DE, partial fraction methods.
Discussion of 2.1-6,16.
Reading on partial fractions [we study (1) sampling, (2) method of atoms, (3) Heaviside cover-up]:
Partial Fraction Theory 2008(125k pdf)
References for 2.1, 2.2, 2.3:
Autonomous DE slides 2008 (pdf)
Newton models, projectile slides 2008 (pdf)
Earth to the moon slides 2008 (pdf)
Verhulst logistic equation (typeset, 5 pages, pdf)
Phase Line and Bifurcation Diagrams (includes "Stability, Funnel, Spout, and bifurcation") (typeset, 6 pages, 161k pdf)
ch2 sections 1,2,3 Slides: 2.1-6,16,38, 2.2-4,10, 2.3-9,27+Escape velocity (8 pages, pdf)
ch2 DEplot maple example 1 for exercises 2.2, 2.3 (1 page, 1k)
ch2 DEplot maple example 2 for exercises 2.2, 2.3 (1 page, 1k)
10 Sep: Collected today, 1.5-8,18,20,34
Lecture on stability theory.
Lecture on 2.1, 2.2 problems. How to construct phase line
diagrams. How to make phase plots. Discussion of 2.2-10,18.
Covered in class for 2.1, 2.2: theory of autonomous DE y'=f(y),
stability, funnel, spout, phase diagram, asymptotic stability, unstable,
equil solution.
Verhulst models with harvesting term.
Work on problems 2.1-6,16 and ask questions on Thursday.
stability theory, phase diagrams,
calculus tools, DE tools, partial fraction methods.
Due soon, Page 106, 2.3: 10, 20
Lecture on midterm 1 problem 5, in conjunction with 2.2-14.
Next: 2.3 and numerical DE topics.
Introduction to Newton models for falling bodies and projectiles.
Newton's laws.
References for 2.1, 2.2, 2.3:
Autonomous DE slides 2008 (pdf)
Newton models, projectile slides 2008 (pdf)
Earth to the moon slides 2008 (pdf)
Verhulst logistic equation (typeset, 5 pages, pdf)
Phase Line and Bifurcation Diagrams (includes "Stability, Funnel, Spout, and bifurcation") (typeset, 6 pages, 161k pdf)
ch2 sections 1,2,3 Slides: 2.1-6, 2.1-16 (rabbit), 2.1-38, 2.2-4, 2.2-10, 2.3-9, 2.3-27+Escape velocity (8 pages, pdf)
ch2 DEplot maple example 1 for exercises 2.2, 2.3 (1 page, 1k)
ch2 DEplot maple example 2 for exercises 2.2, 2.3 (1 page, 1k)
Heaviside partial fraction method (4 pages, 86k)
Heaviside's method and Laplace theory (153k pdf)
Partial Fraction Theory 2008(125k pdf)
11 Sep: Discuss maple 2 in lab. Shiu lectures.
Content [Shiu]:
Maple lab discussed: Lab 2 problem 1. Sources on www. Problems discussed
from 2.1 to 2.3.
Midterm problems 1,2 discussed.
Midterm sample is the S2008 exam, found on the web.
Links for maple lab 2:
maple Lab 2 F2008 (pdf)
maple worksheet text Lab 2 F2008
For more on superposition y=y_p_ + y_h, see Theorem 2 in the link
Linear DE part I (8 pages pdf)
For more about home heating models, read the following link.
Linear equation applications, brine tanks, home heating (typeset, 12 pages, pdf)
12 Sep:
Due today, Page 86, 2.1-6,16. Next time: 2.2-10,14
Free fall with no air resistance F=0.
Linear air resistance models F=kx'.
Week 2, Sep 2,3,5: Sections 1.4, 1.5, 2.1.
02 Sep: Collected in class, Page 26, 1.3-8,14.
Theory of equations review, including quadratic equations, Factor and
root theorem, division algorithm, recovery of the quadratic from its
roots.
The exercises on Page 41, 1.4: 6, 12 will be due next
Friday.
Theory of separable equations continued, section 1.4.
Separation test: F(x)=f(x,y0)/f(x0,y0),
G(y)=f(x0,y), then FG=f if and only if y'=f(x,y) is separable. Basic
theory discussed.
The solutions y=constant are called equilibrium solutions. Find
them using G(c)=0. Non-equilibrium solutions from y'/G(y)=F(x) and a
quadrature step. Implicit and explicit solutions.
Examples for Midterm 1 problem 2.
Example 1: Show that y'=x+y is
not separable using the TEST
Separable Equations slides, separability test, tests I and II (6 pages, pdf)
Example 2: Find the factorization f=F(x)G(y) for y'=f(x,y), given
f(x,y)=2xy+4y+3x+6 [ans: F=x+2, G=2y+3].
Next time 1.5, theory of linear DE y'=-P(x)y+Q(x). Integrating factor,
fraction for replacement of y'+py.
Started topic of partial fractions, to be applied again in 2.1-2.2.
References:
Reference slides for separable DE.
Separable Equations 2007 slides, separability test, tests I and II (6 pages, pdf)
Separable Equations 2008 slides, separability test, tests I and II (9 pages, pdf)
Separable Equations manuscript, classification (pdf)
1.4 Page 40 Exercise slides (4 pages, 500k)
How to do a maple answer check for y'=y+2x (TEXT 1k)
Heaviside coverup method manuscript, 4 pages pdf
Click here
03 Sep:
Evaluation of integrals by the division algorithm. More on the variables
separable method. Solutions for 1.4-6,10.
Linear integrating factor method 1.5. Application to y'+y=e^x. Testing
linear DE y'=f(x,y) by f_y independent of y. Examples of linear equations
and non-linear equations. Picard's theorem implies a linear DE has a unique solution.
Main theorem on linear DE and explicit general solution.
Due next Page 41, 1.4: 6, 10, 18, 22, 26 and Page 54, 1.5: 8, 18, 20, 34.
References for linear DE:
Linear integrating factor method, Section 1.5, slides (pdf)
Linear DE method, Section 1.5 slides: 1.5-3,5,11,33+Brine mixing (9 pages, pdf)
Linear DE part I (Integrating Factor Method), (typeset, 8 pages, pdf)
Linear DE part II (Variation of Parameters, Undetermined Coefficients), (typeset, 7 pages, pdf)
How to do a maple answer check for y'=y+2x (TEXT 1k)
Linear first order slides, integrating factor method (2 pages, pdf)
05 Sep:
Collect in class Page 41, 1.4: 6, 10. Next time Page 41, 1.4: 18, 22, 26
Some solutions for 1.4-6,12,18,22,26.
Linear integrating factor method 1.5. Application to y'+2y=1. Testing
linear DE y'=f(x,y) by f_y independent of y. Examples of linear equations
and non-linear equations. Integrating factor Lemma. Main theorem on linear DE
and explicit general solution.
Three linear examples: y'+(1/x)y=1, y'+y=x, y'+2y=1.
On 1.5-34: The units should be taken as millions of cubic feet.
The textbook gives the initial value problem x'=r_i c_i - (r_0/v)x,
x(0)=x_0. The initial value is x_0 =
(0.25/100)8000, the output rate is r_0=500, and the tank volume is V=8000.
Please determine the value for the
input concentration, constant c_i.
You should obtain r_ic_i=1/4.
Then solve the initial value problem.
The book's answer t = 16 ln 4 = 22.2 days is correct.
Due next, Page 54, 1.5: 8, 18
References for linear DE:
Linear integrating factor method, Section 1.5, slides (pdf)
Linear DE method, Section 1.5 slides: 1.5-3,5,11,33+Brine mixing (9 pages, pdf)
Linear DE part I (Integrating Factor Method), (typeset, 8 pages, pdf)
Linear DE part II (Variation of Parameters, Undetermined Coefficients), (typeset, 7 pages, pdf)
How to do a maple answer check for y'=y+2x (TEXT 1k)
Linear first order slides, integrating factor method (2 pages, pdf)
Reference slides for separable DE.
Separable Equations 2007 slides, separability test, tests I and II (6 pages, pdf)
Separable Equations 2008 slides, separability test, tests I and II (9 pages, pdf)
Separable Equations manuscript, classification (pdf)
Week 1, Aug 25 to 29: Sections 1.1,1.2,1.3,1.4.
25 Aug: Three Fundamental Examples
introduced: growth-decay, Newton Cooling, Verhulst population.
See Three Examples (pdf)
A sliding plate example from mechanical engineering. Preview of modeling,
use of maple and graphics in 2250.
Reference Sliding plates example (pdf)
A cable hoist (elevator) example from mechanical engineering. Preview of 2250 topics.
Reference Cable hoist example (pdf)
Reference MIT cable hoist example from a senior modeling course (pdf)
Background,
3 Examples, Decay Equation Derivation.
Fundamental theorem of calculus. Method of quadrature [integration method
in Edwards-Penney].
Week 1 references (documents, slides)
Three Examples, Fundamental Theorem of Calculus, Method of quadrature, Decay law derivation, Background formulas. 6 slides, pdf.
Three Examples (pdf)
Three Examples, solved 1.2-1,2,5,8,10 by Tyson Black, Jennifer Lahti, GBG, 11 slides, pdf.
Log+exponential Background+Direction fields PDF Document (4 pages, 450k). Decay law derivation. Problem 1.2-2. Direction field examples.
26 Aug:
Collected exercise 1.2-2. Exercises 1.2-4, 1.2-6, 1.2-10 discussed in
class. Slides projected: Tyson Black 1.2-1, Jennifer Lahti 1.2-2,10,
Sections 1.1, 1.2. Example for problem 1.2-1, similar to
1.2-2. Details about exams and dailies.
Panels 1 and 2 in the answer check for an initial value
problem like 1.2-2: y'=(x-2)^2, y(2)=1.
Answer checks. Proof that "0=1" and logic errors in presentations.
Maple tutorials start next week. Maple lab 1 is due soon, please
print it from the link
Maple Lab1.
More on the method of quadrature:
Drill problems and quadrature reference (pdf).
Integration details and how to document them using handwritten
calculations like u-subst, parts, tabular. Maple and Matlab methods.
Integral table methods.
Euler's directional field
visualization, tools for using Euler's idea, reduction of an initial
value problem to infinitely many graphics, showing the behavior of all
solutions, without solving the differential equation.
Lecture on 1.2-8.
Direction field reference:
Direction fields manuscript, 11 pages, pdf.
Threading edge-to-edge solutions
is based upon two rules: (1) Solution curves don't cross, and (2)
Threaded solution curves must match tangents with nearby arrows of the
direction field. See the direction field document above for
explanations.
For problem 1.3-8, xerox at 200 percent the textbook page and paste the
figure. Draw threaded curves on this figure according to the rules in the
direction field document above.
Zoomed copy of Edwards-Penney exercise 1.3-8, to be used to produce HW copy.
27 Aug: Continue lecture on direction fields and existence-uniqueness 1.3. Discuss 1.3-8.
Collected in class Page 16, 1.2: 4, 6.
Integration theory examples. Method of quadrature. Parts, tables, maple.
28 Aug: Intro by Thursday TA staff. Discuss submitted work
presentation ideas. Drill, examples, questions. Discuss problems 1.2-1.3.
29 Aug: Collected nothing.
Discussion of 1.3 problem. Distribution of maple lab 1.
How to thread curves on a direction field. Exercise 1.3-8.
Zoomed copy of Edwards-Penney exercise 1.3-8, to be used to produce HW copy.
Picard-Lindelof Theorem, Peano Theorem, example
y'=(y-2)^(2/3), y(0)=2, like 1.3-14.
The Picard-Lindelof theorem and the Peano theorem are
found in this slide set:
Peano and Picard Theory (3 pages, pdf).
For problem 1.3-14, see
Picard-Lindelof and Peano Existence theory manuscript, 9 pages, pdf
Peano and Picard Theory, 3 slides, pdf
Picard-Lindelof and Peano Existence Example, similar to 1.3-14, 1 slide, pdf
Summary of Peano, Picard and direction Fields [Jan 2008]
Peano, Picard, Direction Fields (slides, pdf)
For 1.3-14, a discussion of background material on functions and
continuity
Click here.
For the write-up of 1.3-14 see part (a) of the link
Picard-Lindelof and Peano Existence Example (1 page, pdf).
See also examples in the summary of Peano, Picard and direction Fields [Jan 2008]
Peano, Picard, Direction Fields (slides, pdf)
Start variables separable DE 1.4.
Classification of y'=f(x,y): quadrature, separable, linear. Venn
diagram of classes. Examples of various types.
Theory of separable equations, section 1.4.
Tests for quadrature (f_y=0) and linear (f_y indep of y) types.
Basic (but useless) theorem: y(x) = H^(-1)( C1 + int(F)),
H(u)=int(1/G,u0..u).
Non-equilibrium solutions from y'/G(y)=F(x) and a
quadrature step.
Variables separable method references:
Reference slides for separable DE.
Separable Equations 2008 slides, separability test, tests I and II (9 pages, pdf)
Separable Equations manuscript, classification (pdf)
1.4 Page 40 Exercise slides (4 pages, 500k)
How to do a maple answer check for y'=y+2x (TEXT 1k)