# 2250 S2008 7:30 and 10:45 Lecture Record

Updated: Wednesday April 23: 09:33AM, 2008     Today: Monday September 24: 11:26AM, 2018
 Week 15: 21 Apr, 23 Apr, 25 Apr. 28 Apr. Week 14: 14 Apr, 16 Apr, 18 Apr. Week 13: 07 Apr, 09 Apr, 11 Apr. Week 12: 31 Mar, 02 Apr, 04 Apr. Week 11: 24 Mar, 26 Mar, 28 Mar. Week 10: 10 Mar, 12 Mar, 14 Mar. Week 9: 03 Mar, 05 Mar, 07 Mar. Week 8: 25 Feb, 27 Feb, 29 Feb. Week 7: 18 Feb, 20 Feb, 22 Feb. Week 6: 11 Feb, 13 Feb, 15 Feb. Week 5: 04 Feb, 06 Feb, 08 Feb. Week 4: 28 Jan, 30 Jan, 01 Feb. Week 3: 22 Jan, 23 Jan, 25 Jan. Week 2: 14 Jan, 16 Jan, 18 Jan. Week 1: 07 Jan, 09 Jan, 11 Jan.

### Week 15, Apr 21,22,23: Sections 7.3, 7.4, exam review

21 Apr: Collected maple Mechanical Oscillations L6.1, L6.2, L6.3. Catch-up day for ch10 problems.
Ch6,ch7,ch10 extra credit due by 4pm on 29 April.
Lecture: Laplace methods and examples for systems. Final exam review ch7 and ch10.
Solving x'=Ax when A is a diagonal matrix [ch1 method] or when A is non-diagonal [ch5 method]. Laplace theory for x'=Ax+F. Resolvent.
Lecture: General systems u'=Au+F and Second order systems
Coupled spring-mass systems of dimension 2 and higher. Railway cars.
Earthquake models.
Complex eigenvalues and how to deal with the expressions for real solutions in the eigenanalysis method for u''+Au=F.

Projected slides:
Newton integral calculus and Laplace calculus. Laplace method. (pdf slides)
Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation. (pdf slides)
Laplace method for systems, resolvent. (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)

22 Apr: Final Exam review. Chapters 5,4,3 in that order.

23 Apr: Last day of class. Collect 7.3, 7.4.
Ch6,Ch7,Ch10 extra credit due by 4pm on 29 April.
Final exam review chapters 10,7,6 in that order.
Correction: "ramp" should be "step' to agree with literature in engineering.
Lecture: Transfer function and convolution.
Lecture: Brine tank models. Recirculating brine tanks.

Projected Slides:
Brine tank cascade. Brine tank recyling, home heating. (pdf slides)
The eigenanalysis method for 2x2 and 3x3 systems x'=Ax. (pdf slides)
Newton integral calculus and Laplace calculus. Laplace method. (pdf slides)
Laplace method for systems, resolvent. (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)
Second order systems, 3x3 spring-mass model, railway cars, earthquakes. Derivation. (pdf slides)

25 Apr: Final Exam day for the 10:45 class, starts at 10:10am in LCB219.

28 Apr: Final Exam day for the 7:30 class, starts at 7:30am in JTB140.

### Week 14: Sections 10.1 to 10.4, Midterm 3

14 Apr:
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].
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
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)

15 Apr: Midterm 3. There are 5 problems.

16 Apr: Collected Page 576, 10.1: 18, 28 and Page 588, 10.2: 10, 16, 24
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.
Partial fractions, Heaviside method, shortcuts, failsafe [sampling] method for partial fractions, method of atoms, 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.
Partial Fractions: How to deal with complex factors like s^2+4. Heaviside's coverup method and how it works in the case of complex roots.
Solving second order DE by Laplace. What to expect and how to do it.
Projected 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)

18 Apr: Collected Page 597, 10.3: 6,18 and Page 606, 10.4: 22
Lecture: Final exam problems.
Convolution theorem. Convolution examples. How to do 10.4 problems.
More on ramp, sawtooth, staircase, rectified sine. Applied expansions of periodic functions as Laplace transforms.
Periodic function theorem. Proof of the periodic function theorem.
Laplace of the square wave, derivation, tanh function.
Laplace of the triangular wave, derivation from the integral theorem.

### Week 13, Apr 7,9,11 Sections 6.2,7.1,7.2,7.3,

07 Apr: Collected 5.6

Exam Review: midterm 3 problem 3: undetermined coefficients, solving homogeneous DE.
Lecture: Eigenanalysis solved problems, sample exam problems. How to use determinants and frame sequences to find eigenpairs and P,D packages.
Lecture: Diagonalization. Equivalence of AP=PD to Fourier's model A(c1 v1 + c2 v2) = c1 lambda1 v1 + c2 lambda2 v2.
Equivalence of AP=PD to the set of equations A v1 = lambda1 v1, A v2 = lambda2 v2.
Eigenvalues from determinants and eigenvectors from frame sequences.
References on Fourier's model, eigenanalysis.
What's eigenanalysis slides 2008 ( pdf)
algebraic eigenanalysis slides 2008 ( pdf)
Eigenanalysis-I manuscript S2008 (typeset 19 pages, 200k pdf)

08 Apr: E. Meucci and L. Zhang, Problems 6.1, 6.2.
Review midterm problems 4, 5 and Fourier's model.

09 Apr: Collected 6.1
Lecture: Complex eigenvalues and eigenvectors. Answer check for eigenpairs [compute AP and PD, then compare AP=PD].
Lecture: Systems of differential equations, position-velocity substitution, conversion of scalar equations to vector-matrix systems, general solution, review problems for exam 3.
Systems of differential equations, conversion of 2x2 scalar linear equations to vector-matrix systems x'=Ax.

Chapter 7 references:
Slides 2008, solving triangular and non-triangular systems (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)

11 Apr: Collected 6.2 and Ch3, Ch4 extra credit.
Lecture: Exam review. Problem 5.
Solving dynamical systems, 2x2 case.
Superposition. Answer check for a general solution. Wronskian test for independence.
main theorem of eigenanalysis and u'=Au.
Solving u'=Au by eigenanalysis. Exercise solutions for sections 7.1, 7.3. Sample solutions for solving u'=Au when A is 2x2, 3x3 and 4x4.
Solving u'=Au in the 2x2 case. Methods 1,2,3,4.
Method 1 is eigenanalysis.
Method 2 is the Chapter 5 method, where x(t) is the solution of a second order equation with characteristic equation det(A- r I)=0. Then y(t) is found by using the first DE, and existing formulas for x(t) and x'(t). This method works when A is not a diagonal matrix.
Method 3 is for diagonal systems u'=Au, in which both differential equations are growth-decay equations solved by Chapter 1 methods.
Method 4 is solving u'=Au+F by Laplace. Details next week.

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 13.
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 12, Mar 31, Apr 2,4: Sections 5.4, 5.6, 6.1, 6.2

31 Mar: Nothing due. Catch-up day.

Problems solved in class: 5.4-20,34 [review] and 5.5-50,62
Lecture 5.4: overdamped, critically damped and under damped behavior, pseudoperiod.
Lecture 5.6: Applications of undetermined coefficients. More fixup rule examples.
Lecture: Pure resonance and practical resonance. Damped forced oscillations. Practical resonance plots.
Lecture 5.6: 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).
Slides projected:
Forced vibrations, undamped case, slides 2008 (pdf)
Forced vibrations, damped case, slides 2008 (pdf)
Forced vibrations and resonance, slides 2008 (pdf)

01 Apr: Meucci and 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).

02 Apr: Collected 5.4-20,34. One stapled package.

Problems solved in class: 5.5: 6, 14, 27, 50, 62 [review] and 5.6: 2, 10, 16 [see web problem notes].
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)
Lecture: One more example of undetermined coefficients. Intro to eigenanalysis. 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)

04 Apr: Collected in class, Page 346, 5.5: 6, 14, 27, 50, 62 in one stapled package. Page 357, 5.6: 2, 10, 16 in one stapled package.
Lecture: How to find the variables lambda and v in Fourier's 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)
Eigenanalysis-I manuscript S2007 (typeset 19 pages, 200k pdf)

### Week 11, Mar 24,26,28: Sections 5.5, 5.4, 5.6

24 Mar: Collect 5.1 problems 34 to 48.

Lecture: Solving constant coefficient nth order DE by finding its list of n atoms. Start section 5.5 undetermined coefficients.

References for weeks 10 and 11.
Picard's Theorem for systems, slides 2008 (pdf)
How to solve linear DE, slides 2008 (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 11 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 1, 2007(pdf)
Undetermined coefficients slides Nov 8, 2007 (pdf)
Ch5 undetermined coeff illustration (3 pages, pdf)

25 Mar: Questions on exam 2 problems 2,3. Questions on 5.2 problems, 5.3-16, 5.3-32.
Midterm 2, problems 2 and 3 [40 min].

26 Mar: Collect Maple L3.2,L3.3,L3.4,L4.2,L4.3,L4.4. Collect 5.2.

Lecture: Questions on maple lab 5. Undetermined coefficients, Related atoms, atomRoot function and the fixup rules 2,3,4. How to find trial solutions quickly. Section 5.5 of Edwards-Penney. A number of slides were projected in class, two from Monday (how to solve the homogeneous equation)
How to solve linear DE, slides 2008 (pdf)
Solving linear DE, examples of orders 1 to 4, slides 2008 (pdf)
and one for today (how to solve for y_p by undetermined coefficients).
Undetermined coefficients slides Nov 1, 2007(pdf)
Undetermined coefficients slides Nov 8, 2007 (pdf)

28 Mar: Collect 5.3 and maple 5. If you could not finish maple labs, then look at the extra credit problems.

Lectur: 5.4 Damped and undamped motion. Pendulum, harmonic oscillations, spring-mass equation, phase-amplitude conversions from the trig course,
Lecture: 5.5 variation of parameters formula (33).
Second order variation of parameters slides 2008 (pdf)
Second order variation of parameters (typeset, 6 pages pdf)

Lecture: 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)

### Week 10, Mar 10,12,14: Sections 4.6, 4.7, 5.1, 5.2, 5.3

10 Mar: Nothing collected.
Sample test for problems 1,4,5.
Lecture: Standard basis in R^n. Theorems on independent sets and bases. Kernel. Nullspace. Image. Column space. Row space. Equivalent bases.
Review independence of functions: sampling test and Wronskian test. Review of 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.
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)

11 Mar: Exam 2 at 7:15 in JTB 140 and 10:30 in LCB 219, proctors Zhang and Meucci. Covers only problems 1,4,5 of exam 2. Problems 2,3 delayed uintil 25 March.

12 Mar: Collected 4.5-8,22,28 and 4.6-2.
Lecture: Proofs involving subspaces for vector spaces V whose data item packages are functions.
Lecture: 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. Picard's Theorem for higher order DE and systems. Dimension of the solution space. Structure of solutions.
Solutions to 4.7-10,22,24.

Week 10 references.
Picard's Theorem for systems, slides 2008 (pdf)
How to solve linear DE, slides 2008 (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)
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

14 mar: Collected 4.7-10,22,24.
Lecture: Second order and higher order differential Equations.
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.
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, Mar 3,5,7: Sections 4.3, 4.4, 4.5, 4.6, 4.7

03 Mar: Collected 4.1-18,22,30. Didn't do 3.6-64? Do the extra credit from ch3.
Extra credit Ch3 due March 12.
Lecture: Independence and dependence. Algebraic tests. Geometric tests. Kernel theorem. Not a subspace theorem. Solutions of problems 4.1, 4.2.
More on the toolkit. Vectors as packages of data items. Examples of vector packaging in applications. The kernel: sols of Ax=0.
Web References:
Lecture slides on Vector spaces, Independence tests. (pdf)
Lecture slides on Basis, Dimension, Rank, kernel, pivot theorem, rowspace, colspace (pdf)

04 Mar: Meucci and Zhang conduct exam 2 review. Questions on 4.3 problems.
Discuss midterm problems 1,4,5.

05 Mar: 4.2 problems due.
Exam review continues from Tuesday: problem 3.
Lecture: More on independence. General solutions with a minimal number of terms. Basis == independence + span. Independence of atoms. Function spaces. Wronskian test. The pivot theorem. Rank test. Determinant test.

07 Mar: Problems 4.3, 4.4 due.
Exam review continues from Tuesday: problems 1,2,3.
Examples in class for Problem 1, midterm 2. To make your own copy of the projected material, click here.
Lecture: Basis == independence + span. Dimension. Basis for linear system Ax=0 from the last frame algorithm. Partial derivatives and bases. The pivot theorem proof. Proof rank(A)=rank(A^T).

### Week 8: Feb 25,27,29: Sections 3.6, 4.1, 4.2, 4.3

25 Feb: Collect Page 194, 3.5: 14, 26, 38.
Lecture: Section 3.6
Determinant slides 2008 (pdf)
Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)
Determinants and Cramer's Rule (typeset 186k pdf)

Methods for computing a determinant. Sarrus' rule, Cofactor expansion, four rules for determinants. In-class examples.
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. Elementary matrices and determinants. Determinant product rule.

26 Feb: Meucci and Zhang: review 3.6 problems, maple lab 2 questions, lecture on maple lab 3 and maple lab 4.
Details on problems 3.6: 6, 18, 30, 38, 64.

27 Feb: Due are Page 194, 3.5: 14, 26, 38 and the last problem 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.
Due today: All chapter 2 extra credit problems, covers 2.1, 2.2, 2.3. Other sections in chapter 2 appear in later extra credit packages.
Lecture: Cofactor expansion. Hybrid methods. 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.
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).
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).

29 FebDue are Page 212, 3.6: 6, 18, 30, 38, 64.
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).
The toolkit of 8 properties (Thm 1, 4.2). Vector spaces. Subspaces. Parking lot analogy. Data recorder example and data conversion to fit physical models. Subspace criterion. Kernel theorem for Ax=0.
Example in class: Apply the subspace theorem to x+y+z=0, writing it as Ax=0, then apply the kernel theorem (thm 2 page 239 of Edwards-Penney).

Lecture: Chapter 4, section 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.

### Week 7, Feb 18,20,22 Sections 3.3, 3.4, 3,5, 3.6

18 Feb: President's Day holiday. No classes.

19 Feb: L. Zhang, review 3.4 problems 20, 30, 34, 40, maple lab 2 questions, maple lab 3 numerical work.

Under construction 20Feb
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)

Report details on 2.4,2.5,2.6 prob 6 (pdf)

20 Feb: Problems 3.3 due.
[Wednesday, 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.
Ch1 extra credit due Friday. Ch2 extra credit due next Wed after that.
Lecture: How to compute the inverse matrix from inverse = adjugate/determinant and also by frame sequences.
Web Reference: Construction of inverses. Theorems on inverses.
slides on rref inverse method S2008

22 Feb: Ch1 extra credit due. Please work on 3.6 and 4.1 problems.
.
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.
Digitial photos and matrices.
Web Reference: Image sensors, digitalphotos, checkerboard analogy, visualization of matrix addition and scalar multiplication.
Digital photos and matrix operations slides S2008
Lecture: Elementary matrices.
Theorem: rref(A)=(product of elementary matrices)A.
Web Reference: Elementary matrices
Elementary matrix slides S2008
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)

### Week 6, Feb 11,13,15: Sections 3.3, 3.4, 3.5

11 Feb: Due today, symbolic sol L3.1, L4.1.
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)
Start of class: Exam 1 review, problems 2,3.
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.

12 Feb, Midterm 1 at 7:10am in JTB 140 or 10:30 in LCB 219.

13 Feb: Page 162, 3.2: 10, 18, 24
In 3,2 solutions, back-substitution should be presented as combo operations in a frame sequence, not as isolated algebraic jibberish.
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].
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.
How to program maple to make a frame sequence without errors.
Review of rref, rank, nullity, dimension with examples.
Review of vector models is in the slide set
Slides on vector models and vector spaces 2008 (pdf)
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 asnwer check [rref(A);].

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)

15 Feb: Due today, Page 170, 3.3: 8, 18 and L1.1, L1.2 the first maple lab. Answer checks should also use the online FAQ.
Problems submitted without frame sequence details are incomplete. Yes, you can use maple to create the sequence and do the answer check.
Lecture: 3.3 and 3.4, Vector form of the solution to a linear system. Matrix multiply and the equation Ax=b.
Slides on matrix Operations (pdf)
Answer to the question: What did I just do, when I found rref(A)?
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).
L2.1 discussed today. Transparencies projected for L2.1 and L2.4 solutions. Happens only in class, no web solutions available.
Due next Week: 3.4-18,30,36,40. 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.

### Week 5, Feb 4,6,8: Sections 2.4,2.5, 2.6, 3.1, 3.2

04 Feb:
Review: study the slides on partial fractions. Due today, Page 106, 2.3: 10, 20
Continue the Friday 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)

05 Feb: Exam 1 review, questions and examples on problems 1,2,3,4,5.
How to present the solutions to L3.1, L4.1.

06Feb: Nothing due today, catch-up day, switching chapters.
Maple lab symbolic sol L3.1, L4.1 will be due Monday. Exam 1 next Tuesday.
Lecture: 3.1, frame sequences, combo, swap, multiply, geometry
Prepare 3.1 problems for Friday.
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)

08 Feb: Due today, Page 152, 3.1: 4, 18, 26
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)

### Week 4, : Sections 2.2, 2.3, 2.4, 2.5

28 Jan: Due today, 1.5-6,18,22,34
Lecture on stability theory. 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)
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 Tuesday.
stability theory, phase diagrams, calculus tools, DE tools, partial fraction methods.
Next: 2.3 and numerical DE topics. Due 05 Feb, Page 106, 2.3: 10, 20
Lecture on midterm 1 problem 5, in conjunction with 2.2-18.
Introduction to Newton models for falling bodies and projectiles.

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)

29 Jan: Discuss maple 2 in lab.
Content [Meucci, Zhang]: Maple labs discussed: 1,2. Sources on www. Problems discussed from 2.1 to 2.3. Midterm problems 3,4 discussed. Midterm sample was distributed in class earlier.
maple Lab 2 S2008 (pdf)
maple worksheet text Lab 2 S2008
For more on superposition y=y_p_ + y_h, see Theorem 2 in the link Linear DE part I (8 pages pdf)
Linear equation applications, brine tanks, home heating (typeset, 12 pages, pdf)

30 Jan:
Due today, Page 86, 2.1-6,16. Next time: 2.2-10,18
Free fall with no air resistance F=0.
Linear air resistance models F=kx'.
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)
Intro to the Jules Verne problem and its solution.
Earth to the moon slides 2008 (pdf)

01 Feb: Due today 2.2-10,18. All of 2.3 and Maple Lab 1 is due on Monday 11 Feb: 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.
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.
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 S2008 (pdf)
Maple L3 snips S2008 (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)
Sample maple code for Euler, Heun, RK4 (maple .mws)
Sample maple code for exact/error reporting (maple .mws)

Report details on 2.4,2.5,2.6 prob 6 (pdf)

### Week 3, Jan 23,25: Sections 1.5, 2.1, 2.2.

22 Jan: Meucci and Zhang: Discussion 1.4 exercises. Questions about maple lab 1. Problem 2 midterm review.

23 Jan: 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

25 Jan: Collected in class, Page 41, 1.4: 18, 22, 26
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, 1.5-33. Some more class discussion of 1.5-34.
Introduction to 2.1, 2.2 topics: autonomous DE, partial fraction methods, Newton's laws.

### Week 2, Jan 14,16,18: Sections 1.4, 1.5, 2.1.

14 Jan: 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.
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.
Variables separable method references:

15 Jan Discussion of 1.3 problem. Distribution of maple lab 1. Exam 1 review, problem 1.
[Meucci and Zhang]: Today in class: All you need to know about quadratics, Start Maple Lab 1.

16 Jan: Collected in class, Page 26, 1.3-8,14. The exercises on Page 41, 1.4: 6, 10 will be due next week. Theory of separable equations continued, section 1.4. Tests for quadrature (f_y=0) and linear (f_y indep of y) types.
Separable equation test.
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].
Basic (but useless) theorem: y(x) = H^(-1)( C1 + int(F)), H(u)=int(1/G,u0..u). 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.
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)

18 Jan:
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.

### Week 1, Jan 7,9,11: Sections 1.1,1.2,1.3.

07 Jan: 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.
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.

08 Jan: Intro by Tuesday TA staff. Discuss submitted work format ideas, examples and problem 1.2-2. Please submit 1.2-2 on Wednesday.

09 Jan: 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, Background, 3 Examples, Decay Equation Derivation.
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.

11 Jan: Continue lecture on direction fields and existence-uniqueness 1.3. Discuss 1.3-8.
Collected in class Page 16, 1.2: 4, 6.
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)