2250 S2007 7:30 and 10:45 Lecture Record

Updated: Monday April 23: 18:50PM, 2007     Today: Friday December 14: 01:05AM, 2018
 Week 15: 23 Apr, 24 Apr, 25 Apr. Week 14: 16 Apr, 18 Apr, 20 Apr. Week 13: 09 Apr, 11 Apr, 13 Apr. Week 12: 02 Apr, 04 Apr, 06 Apr. Week 11: 26 Mar, 28 Mar, 30 Mar. Week 10: 12 Mar, 14 Mar, 16 Mar. Week 9: 05 Mar, 07 Mar, 09 Mar. Week 8: 26 Feb, 28 Feb, 02 Mar. Week 7: 20 Feb, 21 Feb, 23 Feb. Week 6: 12 Feb, 14 Feb, 16 Feb. Week 5: 05 Feb, 07 Feb, 09 Feb. Week 4: 29 Jan, 31 Jan, 02 Feb. Week 3: 22 Jan, 24 Jan, 26 Jan. Week 2: 15 Jan, 17 Jan, 19 Jan. Week 1: 08 Jan, 10 Jan, 12 Jan.

Week 15, Apr 23,24,25: Final Exam Review

23 Apr: Collected Page 597, 10.3: 6, 18
Lecture: Review of final exam chapters 10 and 7, plus one example from chapter 6 on Fourier's model for x'=Ax. 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.

24 Apr: Chen and Todorov: Lecture on chapters 6 and 5, final exam review.

25 Apr: Collected Page 606, 10.4: 22
Lecture: Final exam review, continued, chapters 4 and 3.

Week 14, Apr 16,18,20: Sections 10.1, 10.2, 10.3, 10.4

16 Apr: Collected Page 425, 7.3: 8, 20, 30
Lecture: Exam review. 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.

Laplace theory references week 14.
Ch10 Laplace slides, 10.1 to 10.4 (9 pages, 1.9M pdf)
Laplace theory typeset manuscript (34 pages, 290K pdf)
Heaviside's method, typeset (4 pages, 84K pdf)

17 Apr: Midterm 3, 5 problems

18 Apr: Collected Page 438, 7.4: 6 , Page 576, 10.1: 18, 28
Ch5 extra credit due.
Lecture: Solving differential equations by Laplace's method. Review of rules and table [Laplace calculus]. Partial fractions. Using trig identities [sin 2u = 2 sin u cos u, etc].

20 Apr: Collected maple Mechanical Oscillations L6.1, L6.2, L6.3
Collected Page 588, 10.2: 10, 16, 24
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. Delaying use of the inverse of L to the last step of the solution. How to deal with complex factors like s^2+4. Heaviside's coverup method and how it works. This was the last lecture on LaPlace theory. Next week: Final exam review for 3 days in a row + more LaPlace examples.

Week 13, Apr 9,11,13: Sections 7.1,7.2,7.3,7.4,10.1

09 Apr: Collected Page 370, 6.1: 12, 20, 32, 36

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.

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)

10 Apr: Chen and Todorov, exam 3 review problems 3,4,5. Problems 6.2.
Chen: Midterm 3,4, 5 and Fourier's model.
Todorov:

11 Apr: Collected Page 379, 6.2: 6, 18, 28

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.

13 Apr: Collected Page 400, 7.1: 8, 20
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. Problems from 7.2, 7.3.

Week 12, Apr 2,4,6: Sections 5.6, 6.1, 6.2

02 Apr: Collected 5.4-20,34.

Lecture: Wine glass experiment (7:30). 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: 5.4-34, 5.6-4,8,18.

03 Apr: Chen and Todorov lecture on 5.5 problems and two problems from the F2006 midterm 3 key. They will field questions on 5.6 as well as extra credit problems (ch5).
Todorov: Problems 5.5. Exam 3 - Problems 1 and 2.
Chen: Exam 3, problems 1 and 2, undetermined coefficent and varation of parameter.

04 Apr: Collected in class, Page 346, 5.5: 6, 12, 22.

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)
Eigenanalysis-I manuscript S2007 (typeset 19 pages, 200k pdf)

Slides exist for 6.1 and 6.2 problems. See the web site for the 1.3mb scanned pdf files.

06 Apr: Collected in class, Page 346, 5.5: 54, 58, Page 357, 5.6: 4, 8, 18

Lecture: Eigenanalysis continued. Diagonalization. Differential equations. Eigenvalues from determinants and eigenvectors from frame sequences. Complex eigenvalues and eigenvectors.

Week 11, Mar 26,28,30: Sections 5.5, 5.4, 5.6

26 Mar: Collected 5.1 problems 34 to 48.

Lecture: 5.5 variation of parameters and undetermined coefficients.

Week 11 references
Second order variation of parameters (typeset, 7 pages pdf)
Second order undetermined coefficients (typeset, 8 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)

27 Mar: Chen and todorov lecture on maple lab 5, 5.4-34, 5.3-16, 5.3-32.

Chen and Todorov: Today in class: Maple L5 , 5.3-16 and 5.3-32. Rational roots of polynomials with integer coefficients.

28 Mar: Collect 5.2 and half of 5.3.

Lecture: Undetermined coefficients, fixup rule, related atoms, atomRoot function and the fixup rule. How to find trial solutions quickly. Section 5.5 of Edwards-Penney. Two sets of notes were distributed in class, one for Monday (solve the homogeneous equation) and one for today (solve for y_p by undetermined coefficients).

30 Mar: Collect last half of 5.3 and maple 5. If you could not finish maple 5, then look at the maple 5+6 extra credit in chapter 5 extra credit. That source will appear soon on the web.

Lecture: Sections 5.4, 5.6.
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. Pure resonance and practical resonance. Tacoma narrows bridge. Soldiers in cadence. Damped forced oscillations. Practical resonance plots.

Week 10, Mar 12,14,16: Sections 5.1, 5.2, 5.3, 5.4

12 Mar: Collected 4.4-6,24 and Ch3 extra credit.
Exam review: problems 2,3 again.
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.

Week 10 references.
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)
Week 11 references., after the spring break
Second order variation of parameters (typeset, 7 pages pdf)
Second order undetermined coefficients (typeset, 8 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)

13 Mar: Exam 2 at 7:15 and 10:30 proctors Chen and Todorov.

14 Mar: Collected 4.5-6,24,28 and 4.6-2.
Lecture: Higher order equations.
Atom, independence of atoms, Picard theorem for higher order equations, superposition, solution space structure, dimension of the solution set. Higher order constant equations, homogeneous and non-homogeneous structure. Euler's theorem. Complex roots and exp(i theta)=cos (theta) + i sin(theta) formula. 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.

16 Mar: Collected 4.7-10,20,26 and maple lab 4 [L4.2,L4.3,L4.4]
Exam 2 returned in class.

If you did not finish maple lab 4, then please work on the extra credit maple lab problems in the Ch4 extra credit. Extra credit problems will appear on the web over the break.

Lecture: Problems 5.1,5.2,5.3.
Identifying atoms in linear combinations. Solving more complicated homogeneous equations.

Week 9, Mar 5,7,9: Sections 4.3, 4.4, 4.5, 4.6, 4.7

05 Mar: Collected 4.1 problems and 3.6-60 [you may do the extra credit instead].
Extra credit Ch3 due next Monday, because of snow day March 2.
Lecture: Independence and dependence. Algebraic tests. Geometric tests. General solutions with a minimal number of terms. Basis == independence + span. Standard basis in R^n. Theorems on independent sets and bases. Dimension. Basis for linear system Ax=0.

06 Mar: Chen and Todorov conduct exam 2 review. Also maple 3+4.
Todorov: 2~Induction, 3.6-60 and midterm problems 2,3.
Chen: We did 3.6-60 and then midterm problems 2,3.

07 Mar: 4.2 problems due.
Lecture: Row and column spaces. The pivot theorem. Equivalent bases. 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.

09 Mar: Maple Lab 3 and 4.3 due.
Exam review continues from Tuesday: problem 1 and problem 4.
Lecture: Orthogonality. General vector spaces. Problem session 4.4, 4.5, 4.6, 4.7.

Week 8, Feb 26, 28, Mar 2: Sections 3.6, 4.1, 4.2, 4.3

26 Feb: Collected 3.5 problems. Postpone 3.5-44 until Mar 7. This problem has been replaced by a calculation (no proofs).
Lecture: More examples of Cramer's rule, adjugate formula. Determinant product rule. More determinant theorems. Frame sequences and determinants. Formula for det(A) in terms of swap and mult operations. 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.

27 Feb: Chen and Todorov review 3.6 problems, maple lab 2 questions, lecture on maple lab 3 and maple lab 4.
Todorov: We did something on Lab 3 and 4. Problems 3.6-5, 6, 20, 32.
Chen: Copy text to maple, FAQ, turn in what they have, extra problems, save Maple file, split group and calculate precent error by hands, and modify working maple code.

28 Feb: Maple lab 2 is due. If you are not done, then do the two lab 2 extra credit problems, 7 day limit. See 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.

02 Mar: Due are 3.6-6,20,32,40
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. Due Monday: 4.1.
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 [10:45 only].

Week 7, Feb 20, 21, 23: Sections 3.6, 4.1

20 Feb: Chen and Todorov review 3.4 problems 20, 30, 34, 40, maple lab 2, questions on maple lab 3.

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)

21 Feb: all of 3.4 problems due, 3.4-20,30,34,40
Lecture: 3.5 elementary matrices, 3.6 determinant theory and Cramer's rule.
Distributed in class: xerox on elementary matrices, construction of inverses. Theorems on inverses. Determinants of elementary matrices. Adjugate formula for the inverse. Review of Sarrus' Rules.
Theorem: rref(A)=(product of elementary matrices)A.
Ch3 extra credit available on web 21 Feb.

23 Feb: No problems due. Please work on 3.6 and 4.1 problems.
Lecture: 3.6 determinant theorem for elementary matrices, cofactor expansion, four rules for determinants. Hybrid methods to compute a determinant. Cramer's rule. Adjugate matrix. In-class worksheet on Cramer's rule for 2x2 and on forming adjugates, minors, cofactors. How to compute the inverse matrix from inverse = adjugate/determinant and also by frame sequences. Special theorems for determinants having a zero row, duplicates rows or proportional rows. How to use the 4 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)

References for chapter 4, next week 26 Feb.
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)

Week 6, Feb 12, 14, 16: Sections 3.3, 3.4, 3.5

12 Feb: Due today, Page 162, 3.2: 10, 14, 24.
Lecture: 3.3

13 Feb, Midterm 1

14 Feb: Due today, symbolic sol L3.1, L4.1.
Lecture: 3.3 and 3.4, vector space models
Slides on three possibilities with symbol k,14Feb2007 (3 pages, pdf)

16 Feb: Due today, Page 170, 3.3: 10, 20
Lecture: 3.5 and 3.6

Week 5, Feb 5,7,9: Sections 2.3, 2.5, 2.6, 3.1

05 Feb: Due today, Intro Maple L1.1, L1.2 and Page 106, 2.3: 10, 20
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)

06 Feb: Exam 1 review, problems 2,3,4.

07 Feb: Due today,
07 Feb, postpone symbolic sol L3.1, L4.1 until 14 Feb.
Lecture: 3.1, 3.2, frame sequences and general solution

09 Feb: Due today, Page 152, 3.1: 6, 16, 26
Lecture: 3.2, 3.3, vectors and matrices

Week 4, Jan 29,31, Feb 2: Sections 2.2, 2.3, 2.4,

We are behind the lecture schedule, catching up.

29 Jan: Due today, 2.1-8 only.
Lecture on 2.1, 2.2 problems. How to construct phase line diagrams. How to make phase plots. Problem 5 on midterm 1.
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)

30 Jan: Begin maple 2 in lab.
Content [Chen and Todorov]: Maple labs discussed: 1,2. Sources on www. Problems discussed from 2.1,2.2. Midterm problems 2,3,4 discussed.

31 Jan:
Due today, Page 86, 2.1-16, 2.2-10,14
Discussed midterm problem 4. Lecture on the Jules Verne problem. Information on the reading assignment for 2.3, and the work of Isaac Newton on ascent and descent models for kinematics with air resistance.

02 Feb: 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.
There will be no further discussion of 2.3 in the lectures.
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. 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.
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 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)

Week 3, Jan 22,24,26: Sections 2.1,2.2,2.3,2.4.

22 Jan: Collected in class, Page 41, 1.4: 6, 12. Test for a separable DE. 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 26 Jan, Page 54, 1.5: 8, 18, 20, 34
Reference slides for separable DE test.
Separable Equations slides, separability test, tests I and II (6 pages, pdf)

23 Jan: Chen and Todorov: Review of Exam 1 problems. Discussed 1.5,2.1 exercises. Questions about maple lab 1.

24 Jan: Collected in class, Page 41, 1.4: 18, 22, 26.
Three linear examples: y'=1+y, y'=x+y, y'=x+ (1/x)y. 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 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.
Introduction to 2.1, 2.2 topics. Presentation of 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].

26 Jan: Review partial fraction methods, from Jan 19.
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 and periodic free term. You should be working on problem 2.1-8 and ask questions on 2.1,2.2 problems on Tuesday. Discuss 2.1, 2.2 problems, stability theory, phase diagrams, calculus tools, DE tools, partial fraction methods.
Next week: 2.3 and numerical DE topics. Due 05 Feb, Page 106, 2.3: 10, 20

Week 2, Jan 15,17,19: Sections 1.3,1.4.

15 Jan, Holiday, no classes

16 Jan, More examples of how to write a report, in lab. Discussion of 1.2 and 1.3 problem sets. Distribution of maple lab 1.
[Todorov]: Today in class: All you need to know about quadratics, Maple Lab 1, 1.3-14.
[Chen]: About 40 students attended. We talked about Maple tutorials, Maple lab 1 and questions about the homework.

17 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. Basic (but useless) theorem: y(x) = H^(-1)( C1 + int(F)), H(u)=int(1/G,u0..u). Tests for quadrature (f_y=0) and linear (f_y indep of y) types. Separable equation test delayed to Friday.
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.

19 Jan: Collected in class, Page 26, 1.3: 8, 14. The dailies on Page 41, 1.4: 6, 12 will be due next week. Theory of separable equations continued, section 1.4. Separation test next time: 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.
Started 1.5, theory of linear DE y'=-P(x)y+Q(x). Integrating factor, fraction for replacement of y'+py. Examples next time.
Started topic of partial fractions, to be applied again in 2.1-2.2.
References: see notes for 17 Jan plus
Heaviside coverup method manuscript
PDF Document (4 pages, 86k)

Week 1, Jan 8,10,12: Sections 1.1,1.2,1.3.

Jan 08: Sections 1.1, 1.2. Examples for problems 1.2-1, 1.2-2. Details about exams and dailies. Proof that "0=1" and logic errors in presentations. Panels 1 and 2 in the answer check for an initial value problem like 1.2-2: y'=(x-2)^2, y(2)=1.
"The Three Examples" will be done Friday.
Maple tutorials start next week. Maple lab 1 is due soon, please print it from the web site.

09 Jan: Intro by Chen and Todorov. Discuss submitted work examples and problem 1.2-2.
[Yimin Chen] There are more than 40 students in class today. We talked about the 2 problem sessions we have, homework format and maples. I only talked about 1.2-1,2,4.
[Joro Todorov] There were around 40 students in class today. We discussed the syllabus, the problem format suggestions and then 1.2-2, 6, 10.

10 Jan: Problems 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).

12 Jan: Lecture on direction fields 1.3. Discuss 1.3-8. Three Fundamental Examples introduced: growth-decay, Newton Cooling, Verhulst population. See Three Examples (pdf)
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 (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.