2250 F2007 7:30 and 10:45 Lecture Record
Updated: Wednesday December 05: 17:13PM, 2007
Today: Thursday December 26: 10:09AM, 2024
| Week 15: 03 Dec, |
05 Dec, |
07 Dec, |
| Week 14: 26 Nov, |
28 Nov, |
30 Nov, |
| Week 13: 19 Nov, |
20 Nov, |
21 Nov, |
| Week 12: 12 Nov, |
14 Nov, |
16 Nov, |
| Week 11: 05 Nov, |
07 Nov, |
09 Nov, |
| Week 10: 29 Oct, |
31 Oct, |
02 Nov, |
| Week 9: 22 Oct, |
24 Oct, |
26 Oct, |
| Week 8: 15 Oct, |
17 Oct, |
19 Oct, |
| Week 7: 01 Oct, |
03 Oct, |
05 Oct, |
Week 6: 24 Sep, |
26 Sep, |
26 Sep, |
| Week 5: 17 Sep, |
19 Sep, |
21 Sep, |
| Week 4: 10 Sep, |
12 Sep, |
14 Sep, |
| Week 3: 05 Sep, |
07 Sep, |
| Week 2: 27 Aug, |
29 Aug, |
31 Aug. |
| Week 1: 20 Aug, |
22 Aug, |
24 Aug. |
Week 15, : Sections 10.1 to 10.4, Final Exam Review
03 Dec: Collected Page 576, 10.1: 18, 28 and
Page 588, 10.2: 10, 16, 24
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].
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.
Laplace theory references week 15.
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
(34 pages pdf)
Heaviside's method, typeset
(4 pages, pdf)
04 Dec: Zhang: final exam review.
05 Dec: Collected Maple lab 6. Last day for Ch6 extra credit.
Lecture: Examples of forward and backward table calculations. Harmonic oscillator.
Systems and Cramer's rule. Laplace theory tricks with the Shifting
theorem and the s-differentiation theorem.
Worked problems 10.2 and 10.3 in the 10:45 class.
Projected slides:
Laplace theory examples. Forward and backward table. Shifting, parts, s-differentiation theorems.
(pdf slides)
07 Dec: Collected Page 597, 10.3: 6,18 and Page 606, 10.4: 22
Lecture: Final exam review.
Partial fractions, Heaviside
method, shortcuts, failsafe method for partial fractions, systems of two
differential equations, Cramer's Rule, matrix inversion methods, how to
write up a solution which postpones partial fraction evaluation of
constants to the end. Use of inverse Laplace and Lerch's theorem.
Partial fraction methods for complex roots and roots of multiplicity
higher than one.
How to deal with
complex factors like s^2+4. Heaviside's coverup method and how it works.
Week 14, Nov 26,28,30: Sections 7.3, 7.4, 10.1, 10.2
26 Nov: Collected 6.2 and Page 400, 7.1: 8, 20
Lecture: Exam review. Matrices P,D and Fourier's model as the equation AP=PD.
Diagonalization theory.
Lecture: Answer check for a general solution, Wronskian test
for independence. Second order systems. Complex eigenvalues and
how to deal with the expressions for real solutions in the eigenanalysis
method.
Laplace theory references week 14.
Ch10 Laplace slides, 10.1 to 10.4
(9 pages pdf)
Laplace theory typeset manuscript
(34 pages pdf)
Heaviside's method, typeset
(4 pages, pdf)
27 Nov: Midterm 3, 5 problems
28 Nov: Collected 7.2,7.3.
Lecture: Problems from 7.2, 7.3.
Brine tank models. Recirculating brine tanks. Coupled spring-mass systems of dimension 2.
30 Nov: Collect Monday maple Mechanical
Oscillations L6.1, L6.2, L6.3
Ch5 extra credit due Monday.
Lecture:
Introduction to Laplace's method. The
method of quadrature for higher order equations and systems.
Lerch's theorem. Direct Laplace transform == Laplace integral == L(f(t)).
Shift theorem. Parts theorem. Linearity. The s-differentiation theorem.
Forward table. Backward table. Extensions of the Table.
Week 13, Nov 19,20,21: Sections 6.2,7.1,7.2,7.3,
19 Nov: Collected rest of 5.6 and Page 370, 6.1: 12, 20
Lecture: Eigenanalysis solved problems, sample exam problems.
How to use determinants and frame sequences to find eigenpairs and P,D packages.
Answer check for eigenpairs [compute AP and PD, then compare AP=PD].
Systems of differential equations, conversion of 2x2 scalar linear
equations to vector-matrix systems x'=Ax, solving dynamical systems
(2x2), main theorem of eigenanalysis and x'=Ax.
References on Fourier's model, eigenanalysis.
Fourier's Model slides (5 pages, pdf)
What's eigenanalysis slides ( pdf)
Eigenanalysis-I manuscript S2007 (typeset 19 pages, 200k pdf)
Chapter 7 references:
Slides, solving triangular and non-triangular systems (4 pages, pdf)
Ch7 Systems of DE slides 7.1,7.2,7.3 solved problems (4 pages, 0.8mb pdf)
Recipes for 1st, 2nd order and 2x2 Systems of DE PDF Document (2 pages, 50kb)
DE systems manuscript S2007 (typeset 69 pages, 970k pdf)
20 Nov: L. Zhang, Problems 6.1, 6.2.
Review midterm 3,4, 5 and Fourier's model.
21 Nov: Collected rest of 6.1 and Page 379, 6.2: 6, 18, 28
Lecture: Diagonalization. Differential equations.
Eigenvalues from determinants and eigenvectors from frame sequences.
Complex eigenvalues and eigenvectors.
Lecture: Systems of differential equations, position-velocity
substitution, conversion of scalar equations to vector-matrix systems,
general solution, review problems for exam 3 [Fourier Model, undetermined
coefficients]. Solving x'=Ax when A is a diagonal matrix [ch1 method]
or when A is non-diagonal [ch5 method]. Solving x'=Ax by eigenanalysis.
Week 12, Nov 12,14,16: Sections 5.4, 5.6, 6.1, 6.2
12 Nov: Collected 5.4-20,34.
5.4: Damped and undamped motion. Pendulum, harmonic oscillations,
spring-mass equation, phase-amplitude conversions from the trig course,
overdamped, critically damped and under damped behavior, pseudoperiod.
5.6: Applications of undetermined coefficients. More fixup rule
examples.
Problems solved in class: 5.4-20,34
Slides projected:
Undetermined coefficient example, cafe door, pet door, phase-amplitude, resonance slides F2007
13 Nov: Zhang lecture on 5.4, 5.5 problems and two problems
from the S2007 midterm 3 key. Field questions on 5.4, 5.5, 5.6 as well as
extra credit problems (ch5).
14 Nov: Collected in class, Page 346, 5.5: 6, 12, 22.
Pure resonance and practical resonance. Tacoma narrows bridge.
Soldiers in cadence. Damped forced oscillations. Practical resonance
plots.
Lecture: Wine glass experiment. Tacoma narrows, resonance and vortex shedding.
Soldiers marching in cadence. Theorems on mx''+kx=F0 cos(omega t). Theorems
on mx''+cx'+kx=F0 cos(omega t). Bounded and unbounded solutions. Unique periodic
steady-state solution. Pure resonance omega = sqrt(k/m). Practical resonance
omega = sqrt(k/m - c^2/(2m^2)). Resonance and the fixup rule: omega=sqrt(k/m)
if and only if the fixup rule applies to mx''+kx = F0 cos(omega t).
Problems solved in class: 5.6-4,8,18.
16 Nov: Collected in class, Page 357, 5.6: 4, 8, 18
Lecture: One more example of undetermined coefficients. Intro to eigenanalysis.
Fourier's model. History. How to find the variables lambda and v in Fourier's
model using determinants and frame sequences.
References on Fourier's model, eigenanalysis.
Fourier's Model slides (5 pages, pdf)
What's eigenanalysis slides ( pdf)
Eigenanalysis-I manuscript S2007 (typeset 19 pages, 200k pdf)
Slides exist for 6.1 and 6.2 problems. See the web site.
Week 11, Nov 5,7,9: Sections 5.5, 5.4, 5.6
05 Nov: Collected 5.1 problems 34 to 48.
Lecture: Solving constant coefficient nth order DE by finding its
listof n atoms. Start section 5.5 undetermined coefficients.
Week 11 references
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 coeff, Fixup rules I,II,II (slides, 4 pages pdf)
Ch5 undetermined coeff illustration (3 pages, pdf)
06 Nov: Zhang lecture on maple lab 5, 5.2 problems, 5.3-16, 5.3-32.
07 Nov: Collect 5.2 and half of 5.3.
Lecture: Undetermined coefficients, first fixup rule.
How to find trial solutions
quickly. Section 5.5 of Edwards-Penney. Two sets of slides were
projected in class, one for Monday (solve the homogeneous equation)
and one for today (solve for y_p by undetermined coefficients).
Slides from the classroom notebook computer
How to solve linear constant coefficient nth order DE F2007
How to solve linear nth
order constant coefficient linear homogeneous DE by constructing
an atom list from the characteristic equation. (slides, 2 pages pdf)
Undetermined coefficients, basic trial solution method, F2007
Undetermined coefficient example, cafe door, pet door, phase-amplitude, resonance slides F2007
Week 11 references. Some of these duplicate the notebook slides above.
Atoms and constant-coefficient DE (slides, 5 pages pdf)
Atoms, constant-coefficient DE, examples (slides, 8 pages 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)
09 Nov: Collect last half of 5.3 and maple 4. If you could not finish
maple 4, then look at the extra credit problems.
Lecture: 5.5 variation of parameters formula (33).
Related atoms, atomRoot function and the fixup rules 2,3,4.
Sections 5.4, 5.6.
Week 10, Oct 29,31, Nov 2: Sections 4.6, 4.7, 5.1, 5.2, 5.3
29 Oct: Collected 4.5-6 and maple lab L3.2, L3.2, L3.4.
Exam review: questions answered about exam 2.
Orthogonality. General vector spaces.
Problem session 4.5, 4.6, 4.7.
30 Oct: Exam 2 at 7:15 and 10:30, proctor Zhang.
31 Oct: Collected 4.5-24,28. Delay 4.6-2 until Friday.
Lecture: Proofs involving subspaces for vector spaces V whose
data item packages are functions. Independence of functions: sampling
test and Wronskian test. Review of RREF [RANK] and DETERMINANT test for
independence of fixed vectors. Definition of atom. Independence of atoms.
Method of atoms in partial fractions. Sampling in partial fractions. Heaviside's coverup method.
Solution space theorem for linear differential equations. Solutions to 4.7-10,20,26.
Slides from the classroom notebook computer
Linear equation, frame sequences, general solution slides F2007
The three possibilities with symbol k F2007.
Elementary matrix slides F2007
Vector space, subspace, independence slides F2007
Undetermined coefficients, cafe door, pet door, phase-amplitude, resonance slides F2007
Pivot theorem, column space, row space, equivalence of bases slides F2007
Week 10 references. Some of these duplicate the notebook slides above.
Atoms and constant-coefficient DE (slides, 5 pages pdf)
Atoms, constant-coefficient DE, examples (slides, 8 pages 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)
How to solve linear nth
order constant coefficient linear homogeneous DE by constructing
an atom list from the characteristic equation. (slides, 2 pages pdf)
02 Nov: Collected 4.6-2 and 4.7-10,20,26
Lecture: Second order and higher order differential Equations.
Atom, independence of atoms, Picard theorem for second order
equations, superposition, solution space structure, dimension of the
solution set. Quadratic equations again. Constant-coefficient second
order homogeneous differential equations. Spring-mass DE and RLC-circuit DE
derivations. Electrical-mechanical analogy.
Picard theorem for higher order equations, dimension of the solution
set.
Euler's theorem. Complex roots and the formula exp(i theta)=cos
(theta) + i sin(theta). How to solve homogeneous
equations by searching for a list of n distinct atoms that are solutions
to the equation. Specific examples for first, second and higher order
equations. Common errors in solving higher order equations.
Solved examples like the 5.1,5.2,5.3 problems.
Identifying atoms in linear combinations. Solving more complicated
homogeneous equations.
Higher order constant equations, homogeneous and
non-homogeneous structure. Superposition. Solution space structure.
Week 9, Oct 22,24,26: Sections 4.3, 4.4, 4.5, 4.6, 4.7
22 Oct: Collected 4.1 problems. Didn't do 3.6-60? Do the
extra credit from ch3.
Extra credit Ch3 due next Friday.
Lecture: Independence and dependence. Algebraic tests. Geometric tests.
Kernel theorem. Not a subspace theorem. Solutions of problems 4.1, 4.2.
23 Oct: Zhang conducted exam 2 review. Discussed 4.3 problems.
Discussed two midterm problems.
24 Oct: 4.2 problems due.
Exam review continues from Tuesday: problems 1,2.
Lecture: General solutions with a minimal number of terms. Basis ==
independence + span. Dimension. Basis for linear system Ax=0. Kernel.
Nullspace. Image. Column space. Row space. The pivot theorem.
26 Oct: Problems 4.3, 4.4 due.
Exam review continues from Tuesday: problems 1,3.
Lecture: Standard basis in R^n. Theorems on independent sets and bases.
Equivalent bases.
Example in class for Problem 1, midterm 2. To make your own copy of the
projected material, click here.
Week 8: Oct 15, 17, 19: Sections 3.6, 4.1, 4.2, 4.3
15 Oct: Collect Lab 2 on Tuesday and collect 3.6 problems on Wednesday.
Delay 3.5-44 until Friday.
This problem has been replaced by a calculation (no proofs).
The replacement problem appears as Xc3.5-44a or Xc3.5-44b in the Ch3 extra credit problems,
click here
Review: determinant product rule. More determinant theorems. Frame sequences and
determinants. Formula for det(A) in terms of swap and mult operations.
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.
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).
How to compute determinants of sizes 3x3, 4x4, 5x5 and higher.
Data recorder example and 3.7 theory of least squares.
Intro to ch4.
References for chapter 4.
Vector space, Independence, Basis, Dimension, Rank (typeset, 17 pages, 180k pdf)
Lecture slides on Vector space, Independence, Basis, Dimension, Rank (typeset, 4 pages, pdf)
Lecture slides on Vector spaces, Independence tests. Corrected 20oct06. (typeset, 4 pages, pdf)
Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9 (4 pages, 480k pdf)
16 Oct: Zhang: review 3.6 problems,
maple lab 2 questions, lecture on maple lab 3 and maple lab 4.
Details on problems 3.6-5, 6, 20, 32.
17 Oct: Due 3.6-6,20,32,40,60 and the last 3 problems of Maple lab 2 are due. If you
are not done with lab 2, then do the two lab 2 extra credit problems, 7 day limit.
See the ch3 extra credit problems.
Lecture: Chapter 4, sections 4.1 and 4.2.
Vectors==packages of data items. Parallelogram law. Shadow projection.
Angle theta between vectors. Addition and scalar mult of fixed vectors.
Other vectors: digital photos, Fourier coefficients, Taylor coefficients, sols
to DE like y=2exp(-x^2). The toolkit of 8 properties (Thm 1, 4.2). Vector
spaces. Subspaces. Data recorder example and data conversion to fit
physical models. Subspace criterion. Subspace theorem for Ax=0.
Visualization of matrix addition and scalar multiplication: image sensors
in digital cameras and the checkerboard model. The RGB separation method
of James Clerk Maxwell.
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). (4) Apply the subspace theorem to
x+y+z=0, writing it as Ax=0, then appply thm 2 page 239 of Edwards-Penney.
19 Oct: Due are 4.1-16,20,32 and Xc3.5-44a or Xc3.5-44b in place of 3.5-44.
Lecture: Chapter 4, sections 4.2 and 4.3
More on subspaces: detection of subspaces and data sets that are not
subspaces. Use of theorems 1,2 in section 4.2. Problems 4.1, 4.2 solved
in class.
More on the toolkit. Vectors as packages of data items. Examples
of vector packaging in applications. The kernel: sols of Ax=0.
Intro to independence, dependence.
Week 7, Oct 1, 3, 5
Sections 3.3, 3.4, 3,5, 3.6
01 Oct: Problems 3.3 due.
[Wednesday, all of 3.4 problems will be due, 3.4-20,30,34,40]
Lecture: 3.6 determinant theory and Cramer's rule.
Distributed on the web: PDF on elementary matrices, construction of
inverses. Theorems on inverses.
Adjugate formula for the inverse. Review of Sarrus' Rules.
How to construct the inverse of a matrix using frame sequences.
Discussion of Cayley-Hamilton theorem [3.4-29] and how to solve problem 3.4-30.
Ch1 extra credit due first Wed after fall break. Ch2 extra credit due next Wed after that.
02 Oct: L. Zhang, review 3.4 problems 20, 30, 34, 40, maple lab 2 questions, maple lab 3 numerical work.
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)
03 Oct: Ch1 extra credit due first Wed after fall break. Please work on 3.6 and 4.1 problems.
Due problems 3.4, first two.
Postpone 3.5-44 until after fall break. This problem has been
replaced by a calculation (no proofs) as Xc3.5-44a or Xc3.5-44b.
Lecture: Elementary matrices.
Theorem: rref(A)=(product of elementary matrices)A.
How to compute the inverse matrix from inverse = adjugate/determinant
and also by frame sequences.
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)
05 Oct: 3.4 last two problems due, plus 3.5-16,26.
Lecture: Determinants 3.6.
Problem 3.5-44: Replaced by Xc3.5-44a or Xc3.5-44b, due after Fall Break.
Section 3.6 determinant theorem for elementary matrices,
Determinants of elementary matrices.
cofactor expansion, four rules for determinants.
Hybrid methods to compute a determinant.
Special theorems for determinants having a zero row, duplicates rows or
proportional rows. How to use the 4 rules.
Determinant product rule. More determinant theorems. Frame sequences and
determinants. Formula for det(A) in terms of swap and mult operations.
How to compute determinants of sizes 3x3, 4x4, 5x5 and higher.
Week 6, Sept 24, 26, 28: Sections 3.3, 3.4, 3.5
24 Sep: Due today,
symbolic sol L3.1, L4.1.
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.
The 8-property toolkit for vectors. Parallelogram law. Head-tail rule.
Translation of equation models to (augmented) matrix models and back.
Combo, swap and multiply for matrix models. Frame sequences for matrix
models. Computer algebra systems and error-free frame sequences.
Last 20 minutes: Exam 1 review, problems 3,4.
25 Sept, Midterm 1 at 7:15am in WEB 101 or 10:30 in LS 101.
26 Sep: Nothing due today.
Prepare 3.2 problems 10,14,24 for next time. Please use frame sequences
to display the solution, as in today's lecture examples.
Problems 3.2-10,14: Solve them like the example
Snapshot sequence and general solution, 3x3 system (1 page, pdf, 28-Sep-2006)
In particular, back-substitution should be presented as combo
operations in a frame sequence, not as isolated algebraic jibberish.
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 [page 35 out of 44] in
Linear algebra, no matrices, DRAFT Sept2007 (typeset, 44 pages, pdf)
Lecture: 3.2 and 3.3. How to solve equations in the case of
infinitely many solutions.
Matrices in rref form. Echelon systems. Back-substitution. Detecting
a rref. Gaussian elimination algorithm. Reduced echelon systems and the rref.
Maple packages. Documenting frame sequence steps.
References 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 page, pdf, 28-Sep-2006)
Linear algebra, no matrices, DRAFT Sept2007 (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)
28 Sep: Due today,
Page 162, 3.2: 10, 14, 24.
Due next time, Page 170, 3.3: 10, 20
and L2.1, the first problem of maple lab 2.
Lecture: 3.3 and 3.4, Vector form of the solution to a linear
system. Answer to the question: What did I just do, when I found rref(A)?
L2.1 discussed today. Slides shown for L2.1 and L2.4 solutions.
Due next Wed: all of 3.4.
Week 5, Sep 17,19,21: Sections 2.5, 2.6, 3.1, 3.2, 3.3
17 Sep:
Due today, Page 106, 2.3: 10, 20
For more details on the 2.3 problems,
Click Here.
Continue the Friday 14 Sep lecture on numerical methods.
The problem y'=x+1, y(0)=1 was discussed, including the dot table for
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.
Sample symbolic solution, parallel to L3.1, L4.1 due 24 Sep.
Discussion of Euler, Heun, RK4 algorithms. Computer implementations.
References for numerical methods:
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)
18 sep: Exam 1 review, problems 1,2,3,4,5.
19 Sep: Due today,
symbolic sol L3.1, L4.1 will be due 24 Sep.
Lecture: 3.1, frame sequences, combo, swap, multiply, geometry
References 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 page, pdf, 28-Sep-2006)
Linear algebra, no matrices, DRAFT 18Feb2007(typeset, 37 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)
21 Sep:
Due today, Page 152, 3.1: 6, 16, 26
Lecture: 3.1, 3.2, 3.3, frame sequences, general solution, three possibilities.
The whole hour was spent on exam 1 review, problems 1,2,5.
Week 4, : Sections 2.2, 2.3, 2.4, 2.5
10 Sep: Due today, 1.5-20,34 and 2.1-8,16.
Lecture on 2.1, 2.2 problems. How to construct phase line
diagrams. How to make phase plots.
References for 2.1, 2.2:
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)
11 Sep: Begin maple 2 in lab.
Content [Zhang]:
Maple labs discussed: 1,2. Sources on www. Problems discussed from 2.3.
Midterm problems 2,3 discussed. Midterm sample distributed in class.
Links for maple lab 2:
maple Lab 2 Fall 2007 (pdf)
maple worksheet text Lab 2 Fall 2007
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, 2.2-10,14
Lecture on problems 4, 5, midterm 1.
Free fall with no air resistance F=0.
Linear air resistance models F=kx'.
Information on the reading assignment for 2.3, and the work of Isaac
Newton on ascent and descent models for kinematics with air resistance.
Problem notes for 2.3-10. 2.3-20 are available,
Click Here.
14 Sep: Maple Lab 1 is due: Intro maple L1.1, L1.2.
If you do not turn in this lab, then see the Ch2 Extra Credit problems,
which contains 2 problems like L1.1 and L1.2.
Problem notes for 2.3-10. 2.3-20 are available,
Click Here.
Lecture on the Jules Verne problem from 2.3.
There will be no further discussion of 2.3 in the lectures.
Lectures begin for 2.4, 2.5, 2.6 topics on numerical solutions.
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 1 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)+0, 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 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 Fall 2007 (pdf)
Maple L3 snips Fall 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)
Week 3, Sep 5, 7: Sections 1.5, 2.1, 2.2.
04 Sep: Zhang: Discussed
1.4 exercises. Questions about maple lab 1.
05 Sep: Collected in class, Page 41, 1.4: 6, 12, 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 7 Sep, Page 54, 1.5: 8, 18
References for linear DE:
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 test.
Separable Equations slides, separability test, tests I and II (6 pages, pdf)
07 Sep:
Collected in class, 7 Sep, Page 54, 1.5: 8, 18
Section 1.5, Variation of parameters
formula. General solution of the homogeneous equation. Superposition
principle. Slides for exercises 1.5-3, 1.5-5, 1.5-11, 1.5-33. Some more
discussion of 1.5-34.
Introduction to 2.1, 2.2 topics.
References for 2.1, 2.2:
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)
Week 2, Aug 27, 29, 31: Sections 1.4, 1.5, 2.1.
27 Aug: Collected in class, Page 16, 1.2-10.
Theory of equations review, including quadratic equations, Factor and
root theorem, division algorithm, recovery of the quadratic from its
roots.
Classification of y'=f(x,y): quadrature, separable, linear. Venn
diagram of classes. Examples of various types. Implicit and explicit
definitions. Equilibrium solution. Algorithm for solving a separable
equation.
Tests for quadrature (f_y=0) and linear (f_y indep of y) types.
Separable equation test.
Basic (but useless) theorem: y(x) = H^(-1)( C1 + int(F)),
H(u)=int(1/G,u0..u), delayed until Wednesday.
For 1.3-14, a discussion of background material on functions and
continuity
Click here.
For the write-up of
1.3-14, which is due on Friday, see part (a) of the link
Picard-Lindelof and Peano Existence Example (1 page, pdf).
Start variables separable DE 1.4.
Variables separable method references:
Separable Equations. Separable DE test is here. (typeset, 9 pages, PDF)
1.4 Page 40 Exercise slides (4 pages, 500k)
How to do a maple answer check for y'=y+2x (TEXT 1k)
28 Aug
Discussion of 1.3 and 1.4 problem sets. Distribution of maple lab 1.
[Zhang]: Today in class: All you need to know about quadratics,
Start Maple Lab 1.
29 Aug: Collected in class, Page 26, 1.3-8. Problem 1.3-14
is due Friday. The exercises on Page 41, 1.4: 6, 12 will be due next
week. Theory of separable
equations continued, section 1.4.
Separation test reviewed: 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.
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:
Heaviside coverup method manuscript
PDF Document (4 pages, 86k)
Separable Equations slides, separability test, tests I and II (6 pages, pdf)
31 Aug:
Collect 1.3-14.
Partial fractions continued. Some solutions for 1.4-6,12,18,22,26.
Linear integrating factor method 1.5. Application to y'=2+y. 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.
Due next Page 41, 1.4: 6, 12, 18, 22, 26 and Page 54, 1.5: 8, 18, 20, 34.
References for linear DE:
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)
Week 1, Aug 20,22,24: Sections 1.1,1.2,1.3.
20 Aug: Three Fundamental Examples
introduced: growth-decay, Newton Cooling, Verhulst population.
See Three Examples (pdf)
Fundamental theorem of calculus. Method of quadrature [integration method
in Edwards-Penney].
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.
Maple tutorials start next week. Maple lab 1 is due soon, please
print it from the link
Maple Lab1.
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.
21 Aug: Intro by Tuesday TA staff. Discuss submitted work
examples and problem 1.2-2. Please submit exercise 1.2-2 on Wednesday.
22 Aug: Proof that "0=1" and logic errors in presentations.
Collected exercise 1.2-2. Exercises 1.2-4, 1.2-6, 1.2-10 discussed in
class.
Integration details and how to document them using handwritten
calculations like u-subst, parts, tabular. Maple and Matlab methods.
Integral table methods. Picard-Lindelof Theorem, Peano Theorem, example
y'=(y-2)^(2/3), y(0)=2, like 1.3-14. 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.
The Picard-Lindelof theorem and the Peano theorem are
found in this slide set:
Peano and Picard Theory (3 pages, pdf).
24 Aug: Lecture on direction fields 1.3. Discuss 1.3-8.
Collected in class Page 16, 1.2: 4, 6.
Lecture on Euler's direction field ideas and 1.2-8. Example on y'=(1-y)y
in class.
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.
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