2250 F2006 7:30 and 10:45 Lecture Record

Today: Friday November 24: 20:49PM, 2017
Updated: Sunday November 19: 11:01AM, 2006
Week 13: 13 Nov,  15 Nov,  17 Nov.
Week 12: 06 Nov,  08 Nov,  10 Nov.
Week 11: 30 Oct,  01 Nov,  03 Nov.
Week 10: 23 Oct,  25 Oct,  27 Oct.
Week 9: 16 Oct,  18 Oct,  20 Oct.
Week 8: 09 Oct,  11 Oct,  13 Oct.
Week 7: 02 Oct,  03 Oct,  04 Oct.
Week 6: 25 Sep,  27 Sep,  29 Sep.
Week 5: 18 Sep,  20 Sep,  22 Sep.
Week 4: 11 Sep,  13 Sep,  15 Sep.
Week 3: 05 Sep,  06 Sep,  08 Sep.
Week 2: 28 Aug,  29 Aug,  30 Aug,  01 Sep.
Week 1: 23 Aug,  25 Aug.

Week 13, Nov 13,15,17: Sections 6.1,6.2,7.2,7.3,7.4.





13 Nov:
Monday lecture: Eigenanalysis, differential equations. 6.1,6.2,7.1.

13 Nov Page 357, 5.6: 4, 8, 18
14 Nov, Exam 3 review in lab, continued.
15 Nov Page 370, 6.1: 12, 20, 32, 36
17 Nov Page 379, 6.2: 6, 18, 28


Fourier's Model slides (7 pages, pdf)
Slides, solving triangular and non-triangular systems (4 pages, pdf)

14 Nov [Hwanyong Lee]: Approximately 25 students have shown up today. Lecture: the third midterm exam from Spring 2006. We solved together #1 and #2 and we plan to do #3 next time.



15 Nov:

Lecture: Eigenanalysis, differential equations, algebraic eigenanalysis, three examples, differential equation example. Discuss 6.2 problems.


17 Nov:

Week 12, Nov 6,8,10: Sections 7.2,7.3,7.4.



We are off schedule on the lectures by one week.
This week, 5.6,6.1,6.2,7.1

References week 11-12:
Undetermined coeff, Fixup rules I,II,II (slides, 4 pages pdf)
Atoms, constant-coefficient DE, examples (slides, 8 pages pdf)
Ch5 undetermined coeff illustration (3 pages, pdf)
Undetermined coefficient example, Phase-amplitude conversion, Damping, Resonance, Cafe door, Pet door (12 slides, pdf)


06 Nov:

06 Nov Page 331, 5.4: 20, 34
See Notes on 5.4 problems
Finished undetermined coefficients. Started on 5.6. Review of phase-amplitude form. Example on sum of two harmonic oscillations with differing frequencies.

07 Nov Maple 5 lecture in lab. First 1/4 of Exam 3 review.
[Hwanyong Lee]: Approximately 25 students showed today. I gave a brief summary of 5.1 - 5.3 about three recipes with some examples. And I introduced the free damped motion identifying three cases, overdamped, critically damped and underdamped and we talked about #34 in 5.4.

[Geoffrey Hunter]: About 15 students attended.
I went over some common notation errors I found on the midterms (i.e. what brackets are appropriate for a matrix vs. determinants, clarifying that "Ax" and "b" in the equation Ax=b are both vectors, so it doesn't make sense to write "det(Ax)" because you need a square matrix to calculate a determinant).
5.4-34 has notes on the web, see links below. See also 5.5 notes.


08 Nov:
Lecture: Set of 12 slides above, 4th reference. Undetermined coefficient example. Fixup rule in depth. Started eigenanalysis. Ended ch5 lectures. Ch5 problems were discussed, more Friday. See the note on 5.5 problems linked below also the 5.4 problem like from Nov 6.

08 Nov Page 346, 5.5: 6, 12, 22, 54, 58,
See Notes on 5.5 problems


10 Nov:
Lecture: Algebraic eigenanalysis. Resonance: pure and practical.
to be completed...

Week 11, Oct 30, Nov 1,3: Sections 6.1,6.2,7.1.



We are off schedule on the lectures by one week.

References week 11-12:
Undetermined coeff, Fixup rules I,II,II (slides, 4 pages pdf)
Atoms, constant-coefficient DE, examples (slides, 8 pages pdf)
Ch5 undetermined coeff illustration (3 pages, pdf)


30 Oct:

30 Oct, Exam 2 review continued from Oct 24.
Most of the hour was exam review. Variation of parameters.
Atoms, undetermined coefficient method.

31 Oct, Midterm 2 , LS101


01 Nov:

01 Nov Page 306, 5.2: 18, 22
undetermined coefficients, superposition, nth order eqs
xx Nov Maple Lab 4: Matrices L4.1, L4.2, L4.3 delayed


03 Nov:

03 Nov Page 319, 5.3: 8, 10, 16, 32
Continue undetermined coefficients, existence, recipe nth order eqs

  • Week 10, Oct 23,25,27: Sections 5.4,5.5,5.6.


    23 Oct:

    23 Oct Page 248, 4.3: 18, 24
    23 Oct Page 255, 4.4: 6, 24
    Lecture: Second order recipe.
    Atom, independence of atoms, Picard theorem for second order equations, superposition, solution space structure, dimension of the solution set. Constant equations. Solved 4.1, 4.3, 4.3 problems.

    Week 10 references.
    Atoms and constant-coefficient DE (slides, 5 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)
    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)


    24 Oct:
    24 Oct, Exam 2 review in lab.




    [Geoffrey Hunter]: About 15 students attended.
    Today I went through the answer to question #2 on the sample midterm in detail, then I talked about row and column spaces in 4.5-6.
    I've received a mostly positive response to the Thurs 9:30 tutorial time, so we'll stick to that time.

    [Hwanyong Lee]: About 30 students attended.
    We talked about exam problems 1,2,3 of the Spring 2006 exam.

    25 Oct:

    25 Oct Page 263, 4.5: 6, 24, 28
    25 Oct Page 271, 4.6: 2,
    Lecture: Review day for problems in chapter 4. Worked assigned exercises in 4.4, 4.5, 4.6, 4.7. More on subspaces, independence, wronskian test, sample test for functions (see pdf week 10 references).


    27 Oct:
    27 Oct Page 278, 4.7: 10, 20, 26
    27 Oct Page 294, 5.1: 34, 36, 38, 40, 42, 46, 48

    Lecture: Exam problem 4 at 10:45. At 7:30, no exam problems, delayed until Monday.
    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 recipes for first, second and higher order equations. Common errors in solving higher order equations.
    Monday: Introduction to undetermined coefficients, solving more complicated homogeneous equations, method of variation of parameters.

    Week 9, Oct 16,18,20: Sections 4.7,5.1,5.2,5.3.



    16 Oct:
    We are still behind but catching up. Currently on 4.3 at the end of Monday 16 Oct. New collection schedule: see the due dates page.
    Due 16 Oct Page 194, 3.5: 16, 26, 44
    Lecture on problems 4.1, vector spaces, basis, dimension, rank, nullity, pivot theorem, pivot method, rowspace, colspace, nullspace.
    Web references for Ch4.
    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)
    Three rules, frame sequence, maple syntax (typeset, 7 pages, pdf, 12 Oct 2006)



    17 Oct:

    Hunter and Lee talked about 3.4 to 4.2 problems and maple lab 4.
    [Hwanyong Lee]: Today approximately 30 students showed up. We talked about the definition of a vector space and a subspace. And we discussed the THM on subspaces and Av=0 plus an example taken from problems 29 to 32 in 4.1. We talked about a linear combination and problem #9.

    [Geoffrey Hunter]: Tues: About20 students.
    I reintroduced what a basis, vector space, and vector subspace are and gave the example of v = (x y z)^T being a vector space (in R3) and w = (x 0 z)^T being a subspace of V. I also showed that r = (x y z)^T with y>0 is not a subspace.
    I also clarified what it meant for the statements in Theorem 7 (section 3.6) to be "equivalent".
    Lastly, we discussed what induction is, when to use it, and the three basic steps of induction, i.e. 1. Base case proof where you prove the statement for n=0 (or the "smallest applicable n") 2. Assume the statement is true for all n <=k 3. Prove the statement for n = k+1
    Maple 4 lab was started, but not finished yet. More in later labs.


    18 Oct:
    Due 18 Oct Page 212, 3.6: 6, 20, 32, 40, 60

    Collection of 3.6-32, 3.6-40, 3.6-60 is subject to REDO, so you may submit correctioned versions if the score is <90 percent. The Wed lecture was on indepemdence test RANK + DETERMINANT as documented in the slides listed above for 16 Oct. Covered 4.1, 4.2, 4.3 problems. Subspaces discussed. Subspace test, subspace theorem with Ax=0. How to tell if a set is not a subspace. More on Frame Sequences, rref, general solution in vector form, basis, rank, nullity.


    20 Oct:
    Due 20 Oct Page 233, 4.1: 16, 20, 32 [discussed 16 Oct]
    Due 20 Oct Page 240, 4.2: 4, 18, 28


    Due 23 Oct Page 248, 4.3: 18, 24
    Due 23 Oct Page 255, 4.4: 6, 24

    Week 8, Oct 9,11,13: Sections 4.3,4.4,4.5,4.6.




    09 Oct:
    09 Oct Page 194, 3.5: 16, 26, 44 will be due 11 Oct. Maple labs 2 and 3 will be collected over the next 10 days.

    Lecture on determinants, 3.6. Cramer's rule, Determinant product theorem.
    The 4 rules to compute any determinant: triangular, swap, combo, mult. Cofactor expansion for 3x3, 4x4 and nxn determinants. Elementary matrices. Main theorem: det(A)=(-1)^s/m where s=number of el swap matrices and m = product of the multipliers for the el multiply matrices, in a frame sequence taking A to rref(A). Determinant rules: zero row implies zero det; proportional rows implies zero det. Cramers rule statement 2x2 and nxn. Adjugate inverse formula statement.

    Week 8 slides
    Elementary matrices, vector spaces, slides (8 pages, pdf, 12 Oct 2006)
    Three rules, frame sequence, maple syntax (typeset, 8 pages, pdf, 12 Oct 2006)
    Determinants and Cramer's Rule (typeset, 11 pages, 140k pdf)
    Vector space, Independence, Basis, Dimension, Rank (typeset, 17 pages, 180k pdf)
    Ch4 Page 237+ slides, Exercises 4.1 to 4.4, some 4.9 (4 pages, 480k pdf)


    10 Oct:
    10 Oct Lab [Hunter and Lee, EMCB 101, LS 101]: Begin maple 4 Matrices, Ch3 problem session.

    [Hwanyong Lee]: Today approximately 38 students showed up. I distributed the Maple lab 4. We talked about the Maple lab 2, L2.1, which is due 11 Oct,like how to find u_h in prob. 1 and how to find the steady state solution in pro. 2 etc. And we talked about #44 comparing with theorem 6.

    [Geoffrey Hunter] Tues: 20 students
    I gave the students a starting point with Question 3.4-30 and briefly reviewed how to do matrix-vector and matrix-matrix multiplication. I also showed them how to enter a function with more than 1 argument into Maple and helped them with some of the Maple 2 lab.


    11 Oct:
    11 Oct Page 212, 3.6: 6, 20, 32, 40, 60 will be due on 13 Oct.
    Slide on lab 2 problem 4. Explain graphics.


    13 Oct:
    13 Oct Page 233, 4.1: 16, 20, 32 will be due Monday 16 Oct.
    Discussed 3.6 problems in class, started on vector spaces, independence, dependence. Algebraic criteria, geometric examples.

    Week 7, Oct 2,4: Sections 4.1,4.2.

    We are still behind, currently on 3.4-3.5.

    02 Oct:
    02 Oct Exam 1 review continued: problems 3,4.
    Elementary matrices. Thm: rref(A1)=E_k ... E_2 E_1 A1, a matrix A1 (frame 1) has RREF equal to the product of elementary matrices times A1. Each elementary matrix corresponds to the swap, combo or mult operation present in a frame of the sequence taking A1 to the last frame rref(A1). Inverses. Definition, uniqueness, formula for 2x2 case. Inverse answer check: AB=I does it, because by a theorem, this implies the other relation BA=I.

    Week 7 slides:
    Linear equations, reduced echelon, three rules (typeset slides, 7 pages, pdf)
    Snapshot frame sequence and general solution, 3x3 system (1 page, pdf)
    Vectors and matrices (typeset, 11 pages, 113k, pdf)
    Matrix equations (typeset, 6 pages, 92k, pdf)
    Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)


    03 Oct:
    03 Oct Midterm 1


    04 Oct:
    04 Oct Page 182, 3.4: 20, 30, 34, 40
    Inverses of elementary matrices, how to form them without computation. An inverse of any nxn matrix A is the product of elementary matrices. More on the RREF. How to detect an rref. Pivot columns. More on inverses and computation. Inversion by adjugate matrix and determinant theory (to be finished in 3.6). Inversion of A by augmentation of the identity: let C=aug(A,I), compute a frame sequence to rref(C), and read off the inverse of A from the right panel. Proof of this fact in the 2x2 case, and how to generalize the proof to nxn matrices. Examples from 3.4, 3.5 problems.

    10:45 session: started determinant theory. College algebra definition, Sarrus' rule, cofactor expansion definition.

    05-06 Oct Fall Break

    Week 6, Sep 25,27,29: Sections 3.5,3.6.

    We are still behind, currently on 3.2 for Monday 25 Sept.

    25 Sep:
    Due 25 Sep Page 152, 3.1: 6, 16, 26
    Slides shown in class for maple calculations in MAPLE lab 3 problems L3.1 and L3.3. These were made by modifying the following sources:
    Maple Worksheet files: Press SHIFT and then mouse-click to save the file to disk. Open the saved file in xmaple. In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...".
    Sample maple code for Euler, Heun, RK4 (maple worksheet)
    Sample maple code for exact/error reporting (maple worksheet)
    Covering 3.2: reduced echelon system, lead variables, free variables, general solution, snapshot sequence, three possibilities, 3 rules (combo, swap, mult). The role of variable list order in the RREF method (== elimination). Problems from 3.1 and 3.2 [slides on web].
  • Weeks 5,6 slides
    Linear equations, reduced echelon, three rules (typeset, 7 pages, pdf)
    Snapshot sequence and general solution, 3x3 system (1 page, pdf)
    Linear algebra, no matrices, (typeset, 21 pages, pdf)
    Vectors and matrices (11 pages, 113k, pdf)
    Matrix equations (typeset, 6 pages, 92k, pdf)
    Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)
    26 Sep:
    Exam 1 review in lab, problem 1. Discussion of maple lab 3, L3.1 and L3.2 problems. Problems 3.1 and 3.2 by request.
    [Goeffrey Hunter]: Tues: 20 students, Wed: 8 students
    Tues: I did an overview of the steps involved in solving ODEs with an integrating factor vs. separation of variables (i.e. how do you know which method to use to solve an ODE? Why do you use the integrating factor?) and commented on some observations that would make computations faster. I also reviewed the structure of a Maple for loop.
    Wed: After deflecting numerous what's on the test questions, we went through the sample test that was on the site. Integration of int x^2/(1+x) dx (see exam problem 3) tabled until Thursday class meeting.
    [Hwanyong Lee]: About 30 students have showed up at today's class. We talked about how to get maple code from the web page. And we talked about the Maple Project 3 and how to copy the maple code from the web page to the work sheet in xmaple. I distributed sample Midterm 1 (old Spring 2006) and we solved it together.

    27 Sep:
    Due 27 Sep, Page 162, 3.2: 10, 14, 24 Did sample integrations for problem 1, midterm 1.
    Started 3.2: reduced echelon system, lead variables, free variables, general solution, snapshot sequence, three possibilities, 3 rules (combo, swap, mult). The role of variable list order in the RREF method (== elimination). Problems from 3.1 and 3.2 [slides on web].

    29 Sep:
    Due 29 Sep Page 170, 3.3: 10, 20
    Due 29 Sep, Maple 3, Numerical DE problem L3.1, 200 possible. Problems L3.2 to L3.6 due after the break. Slides shown in class 25 Sept for maple calculation in L3.1. The slides are not on web, but the source code is there in an example:
    Sample maple code for Euler, Heun, RK4 (maple worksheet)
    Use this code to start, get it to work, then edit it for maple lab L3.1.
    Typo discovered in L3.1: (x=2.1) should be (x=0.25 or 0.1)
    Finished 3.2: reduced echelon system, lead variables, free variables, general solution, snapshot sequence, three possibilities, 3 rules (combo, swap, mult). The role of variable list order in the RREF method (== elimination). Problems from 3.1 and 3.2 [slides on web].
    All of 3.3 covered plus problems 3.3 and started on 3.4-20,30,40 (34 has web notes). More on the snapshot frame sequence, rref, maple commands to find the frame sequence.
    Exam Review: Did classification theory for problem 2, midterm 1. Showed how to test about 8 equations, for type quadrature, linear or separable. Showed why y'=x+y is not separable.

    Week 5, Sep 18,20,22: Sections 2.5,2.6,3.1.


    We are behind the lecture schedule, catching up. This was due to missing about 1 day and getting behind about 1 day. Partial catch-up by skipping 2.3.


    18 Sep
    7:30 class: Review of RECT, TRAP, SIMP rules. Introduction to numerical solutions of y'=f(x,y), y(x0)=y0. Euler, Heun and RK4 algorithms. Predictor-corrector methods. Motivation for the algorithms. Reduction of the algorithms to RECT, TRAP and SIMP in the case of a quadrature DE. Explanation of computer algorithms in maple. Comparison of codes for RECT, TRAP, EULER, HEUN.
    10:45 class: Review of RECT, TRAP, SIMP rules. Introduction to numerical solutions of quadrature problems y'=F(x), y(x0)=y0. Four examples again, from last time, including symbolic solutions to 2.4-6 and 2.4-12: y'=-2xy, y(0)=2, y=2exp(-x^2) and y'=(1/2)(y-1)^2, y(0)=2, y=(x-4)/(x-2). Also included: 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). Did the dot table for x=0, 0.1, 0.2, 0.3 and y= 2, 2.1, 2.2 ,2.3.
    Reference:
    Slides on Rect,Trap,Simp,Euler,Heun,RK4 (39k pdf)

    19 Sep

    [Hwanyong Lee]: Approximately 25 students showed up at today's class. We talked about the Maple project 2 , in particular Prob 1 and 2. Discussed was problem 2.4-12, how to find y_n and how to solve explicitly for the symbolic solution. Also discussed was the solution of y'=exp(x^2) and the rectangular, trapezoidal and simpson rules which are used to approximate the integral of exp(x^2). [Goeffrey Hunter] Tues Morning Tutorial: Approx 25 students on Tues showed up. I announced the information on the web site for the test review and that the students should review the material before the in class review sessions. I also reannounced that Maple Lab #2 is on-line.
    We reviewed Euler's method and explained that it was a linear approximation to the actual derivative (i.e. f(x,y) ~ rise/run = (y_(n+1) - y_n)/h ). We talked about question 2.4-6 and how to use Euler's method to approximate y(0.5) and we reviewed how to obtain the symbolic answer using separation of variables. The symbolic solution details and answer check were discussed.

    [Goeffrey Hunter] maple code for y'=2xy, y(0)=2.


    > # This will compute a numerical solution to an ODE using Euler's method. 
    > # Only Step 1-3 information needs to be changed for different problems.
    > #
    > # Step 1: Provide initial data
    > y0 := 2:
    > x0 := 0:
    > 
    > # Step 2: Define ODE 
    > f := (x,y) -> 2*x*y:
    > 
    > # Step 3: Stepsize and number of nodes
    > h := 0.5:
    > n := 5:
    > 
    > # The remainder of this code need not be changed. 
    > # Store data in 2 arrays
    > Y := array(1..n):
    > X := array(1..n):
    > # Initialize first element in array
    > Y[1] := y0:
    > X[1] := x0:
    > # Compute the solution using Euler's method      
    > for i from 1 to n-1 do
    > X[i+1] := x0 + h*i;
    > Y[i+1] := Y[i] + h*f(X[i],Y[i]);
    > end do:
    > # Define a list to plot the information in.
    > l := [[X[r],Y[r]] $r=1..n]:
    > # Plot the solution
    > plot(l,x=X[0]..X[n]);
    <\pre>
    
    
    20 Sep

    Introduction to numerical solutions of y'=f(x,y), y(x0)=y0. Euler, Heun and RK4 algorithms. Predictor-corrector methods. Motivation for the algorithms. Reduction of the algorithms to RECT, TRAP and SIMP in the case of a quadrature DE. Explanation of computer algorithms in maple. Comparison of codes for RECT, TRAP, EULER, HEUN.
    Four examples again, from last time, including symbolic solutions to 2.4-6 and 2.4-12: y'=-2xy, y(0)=2, y=2exp(-x^2) and y'=(1/2)(y-1)^2, y(0)=2, y=(x-4)/(x-2). Also included: 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). Dot table for x=0, 0.1, 0.2, 0.3 and y= 2, 2.1, 2.2 ,2.3.
    Worked in class: y'=2xy, y(0)=2 by Euler and Heun methods. Showed how to use dummy variable names x0, y0 to fill out the rows of the dot table using a single formula (Euler, Heun).
    Introduction to linear algebra. Systems in variables x,y. The three possibilities. Cramer's rule and elimination. Signal equation "0=1". Scope of our study of linear algebra: we stop at eigenanalysis.

    References for numerical methods:
    Slides on Rect,Trap,Simp,Euler,Heun,RK4 (39k pdf)
    Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
    Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)
  • Introduction to Maple Lab 3, Numerical Solutions
    Maple Lab 3 Fall 2006 (pdf)
    The actual symbolic solution derivation and answer check are submitted as dailies, separately. See the due dates page. A sample derivation and answer check appears in the sample symbolic solution report for 2.4-3 below.
    Report details on 2.4,2.5,2.6 prob 6 (pdf)
    Report details on 2.4,2.5,2.6 prob 12 (pdf)
    Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
    Sample symbolic solution report for 2.4-3 (pdf 1 page, 120k)
    Sample Report for 2.4-3 (pdf 3 pages, 350k. Page 1 is symbolic sol.)
    Numerical DE coding hints, TEXT Document (1 pages, 2k)
    ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)
    Maple Worksheet files: Press SHIFT and then mouse-click to save the file to disk. Open the saved file in xmaple. In Mozilla firefox, save to disk using right-mouseclick and then "Save link as...".
    Sample maple code for Euler, Heun, RK4 (maple worksheet)
    Sample maple code for exact/error reporting (maple worksheet)
    22 Sep

    Continued linear algebra and simultaneous linear equations. Reduced echelon system. Cramer's rule and the three possibilities: (1) No sol, (2) Infinitely many sols, (3) Unique solution. The result: determinant not zero if and only if (3); determinant zero if and only (1) or (2). The three rules for elimination. Snapshot sequence example 2x2. Frames. First frame==original system, Last Frame==reduced echelon system. Logic: in the unique solution case (3), the reduced echelon system is a list of equations in variable list order, which assigns to each variable a unique number. The list has this essential property: each nonzero equation has a leading variable, i.e., a variable that appears just once in the whole list, and it appears first, read left-to-right, with coefficient 1. How to write the general solution in the infinitely many solution case (2). Signal equation and no solution, which means no equations for the variables and no answer check (there is no answer to check!). Due Monday: 3.1 problems.
  • Weeks 5,6 slides
    Linear equations, reduced echelon, three rules (typeset, 7 pages, pdf)
    Snapshot sequence and general solution, 3x3 system (1 page, pdf)
    Linear algebra, no matrices, (typeset, 21 pages, pdf)
    PDF Document (11 pages, 113k)
    Matrix equations (typeset, 6 pages, 92k)
    Ch3 Page 149+ slides, Exercises 3.1 to 3.6 (6 pages, 890k pdf)

    Week 4, Sep 11,13,15: Sections 2.5,2.6,3.1.


    We are behind the lecture schedule, catching up.

    On 1.5-34: there was a lecture in class. You were left to discover the input concentration constant c_i, and solve 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, using units of millions of cubic feet. 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 problems on Tuesday.
    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)

    12 Sep, Begin maple 2 in lab.
    Content [Geoffrey Hunter]: We discussed question 1.5-34 in both sections in great detail. Started with the dp/dt = stuff in - stuff out equation and derived the full DE, partly intuitive. Discussed units and ways to check units, a sanity check. Solved 2.1-15. Reviewed partial fractions, applied theory to a problem. Both sections also wanted further clarification as to the purpose of the integrating factor and when to use it. On Tues, I showed two different ways that 1.5-20 could be solved (integrating factor and separation of variables), while on Wed I just went through question 1.5-18. Maple labs discussed: 1,2. Sources on www.
    Content [Hwanyong Lee]: I distributed maple lab2. We discussed about maple lab 1, like how to control the size of plots and how to solve etc. Discussed how to find the general solution of the linear first-order equation. Solved problem 1.5-18. In addition, we discussed about the existence of the unique solution of the linear first-order equation by looking at Peano Thm and Picard-Lindelof Thm. Sec.2.1, 2.2 next time, if at all.


    Due 15 Sep, Page 86, 2.1: 8, 16
    7:30 class: discussed 2.2 problems, stability theory, phase diagrams, calculus tools, DE tools, partial fraction methods. 10:45 class: Class ended after 5 minutes. As a result, 2.3 will be skipped.
  • Heaviside coverup method manuscript
    PDF Document (4 pages, 86k)

    15 Sep Maple Lab 1 is due: Intro maple L1.1, L1.2. Due 18 Sep, Page 96, 2.2: 10, 14.
    Never due: Page 106, 2.3: 10, 20. It will be extra credit, for Dec 7.
    7:30 class: continued with partial fractions and introduction to numerical solutions of quadrature problems y'=F(x), y(x0)=y0. Discussed 2.2 problems. Four examples, including symbolic solutions to 2.4-6 and 2.4-12: y'=-2xy, y(0)=2, y=2exp(-x^2) and y'=(1/2)(y-1)^2, y(0)=2, y=(x-4)/(x-2). Also included: 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). Did the dot table for x=0, 0.1, 0.2, 0.3 and y= 2, 2.1, 2.2 ,2.3. 10:45 class: discussed 2.2 problems, stability theory, phase diagrams, calculus tools, DE tools, partial fraction methods. Some work on RECT, TRAP rules.
    References for numerical methods:
    ch2 Numerical Methods Slides, Exercises 2.4-5,2.5-5,2.6-5 plus Rect, Trap, Simp rules (5 pages, pdf)
    Report details on 2.4,2.5,2.6 prob 6
    Report details on 2.4,2.5,2.6 prob 12
    Numerical Solution of First Order DE (typeset, 19 pages, 220k pdf)
    Sample Report for 2.4-3 (pdf 3 pages, 350k)
    Numerical DE coding hints, TEXT Document (1 pages, 2k)
    Sample maple code for Euler, Heun, RK4 (maple worksheet)
    Sample maple code for exact/error reporting (maple worksheet)

    Week 3, Sep 4,6,8: Sections 2.1,2.2,2.3,2.4.


    04 Sep: Holiday


    05 Sep: Begin maple 1 in lab, Hunter and Lee. Discussed exercises section 1.4. Trouble with room ST 104. Go to LS 101 for the rest of the semester.
    [Hwanyong Lee]: We discussed the maple lab1, due in about one week; see the web due dates for the exact moment it is due. Discussed was the separable D.E. is and #2 as its example. We talked about a D.E. which is not separable, the definitions of implicit solution and explicit solution and the difference between them. Then we discussed how to solve y'=1+y and what meaning the solution y=-1 has comparing with a general solution.
    [Geoffrey Hunter]: Same as H. Lee, plus remarks on question 1.4-6. I was asked about how to check if a solution was correct, a logic question. I used that question to explain skipped solutions. Attendance: 25 at 7:30 Tuesday, 5 at 5:55 Wednesday.


    06 Sep: Collected in class, Page 41, 1.4: 6, 12. Review of Picard-Lindelof and Peano theorems. More on problem 1.3-14. Theory of variables separable equations. Solutions to 1.4-6,12,18,22,26. Linear integrating factor method 1.5. Application to y'=1+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.


    08 Sep: Collected in class, Page 41, 1.4: 18, 22, 26. Due Monday, 11 Sep, Page 54, 1.5: 8, 18, 20, 34 Three 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. Introduction to 2.1, 2.2 topics. Presentation of Midterm 1 problems 1 to 4. Started topic of partial fractions, 2.1-2.2. 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)

    Week 2, Aug 28,30, Sep 1: Sections 1.3,1.4,1.5.


    28 Aug: Collected in class Page 16, 1.2: 4, 6, 10. Lecture on Euler's direction field ideas. Example on y'=(1-y)y in class. See Three Examples (pdf) Projected slide from page 1 of the document Direction fields (11 pages, pdf). Threading edge-to-edge solutions was 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. Also stated in class was the Picard-Lindelof theorem, found in this slide set: Peano and Picard Theory (3 pages, pdf). 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 (typeset, 9 pages, pdf) and
    Peano and Picard Theory (3 pages, pdf) and
    Picard-Lindelof and Peano Existence Example, similar to 1.3-14 (1 page, pdf)


    29 Aug, More examples of how to write a report, in lab. [from Hwanyong Lee] We discussed briefly the Format Suggestions for submitted work. For Sec 1.1, we discussed the definition of the differential equation and the difference between the ordinary differential equation and the partial differential equation. For Sec 1.2, we discussed how to find the solution of y'=f(x) and why we can find the solution explictly. we solved together problems #1, #6. For Sec 1.3, we discussed why we can't use the method used in the previous section to find the solution of y'=f(x,y) and we discussed the Peano theorem and the Picard Lindelof theorem and the difference between the two theorems. And we discussed together the problems #13, #14 and the difference between two problems.
    [Geoffrey Hunter] We discussed the format requirements for assignments. Referenced the sample work on the web site and referred specific questions about format to MWF classes. I clarified some points of confusion regarding the time/location of the Maple labs and reminded the Wed class of the Maple lab tutorial notice sent by e-mail. I answered questions on 1.3-14 (Tues and Wed) and did 1.3-11 to reinforce the ideas (Wed). I also did a verbal comparison of 1.3-14 with 1.3-13 (Tues and Wed). Briefly discussed (with 1.3-14) how you can have existence but not necessarily uniqueness of solutions.


    30 Aug: Collected Page 26, 1.3: 8 in class. Discussion of 1.3-8 graphic. More on threading edge-to-edge solutions and Rules 1,2. Rule 1: Solution curves don't cross, and Rule 2: Threaded solution curves must match tangents with nearby arrows of the direction field. See the direction field document above for explanations. More on Picard-Lindelof and Peano theorems. See Peano and Picard Theory (3 pages, pdf) for the statement of results. For a discussion of background material on functions and continuity, Click here. For the discussion of 1.3-14, which is due Friday 1 Sep, see part (a) of the link Picard-Lindelof and Peano Existence Example (1 page, pdf). Started in the 10:45 class only is section 1.4, on variables separable DE. 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)


    01 Sep: Collected in class, Page 26, 1.3: 14. The dailies on Page 41, 1.4: 6, 12 will be due Wednesday 6 Sep. Theory of separable equations started, section 1.4. Separation test: F(x)=f(x,y0)/f(x0,y0), G(y)=f(x0,y), then FG=f if and only if y'=f(x,y) is separable. Basic theory discussed. Venn diagram of first order equations of type quadrature, separable and linear. Tests for quadrature (f_y=0) and linear (f_y indep of y) types. Skipped solutions y=constant and how to find them using G(c)=0. Non-equilibrium solutions from y'/G(y)=F(x) and a quadrature step. Implicit and explicit solutions. References: see 30 Aug notes.

    Week 1, Aug 23,25: Sections 1.1,1.2.


    Aug 23: Sections 1.1, 1.2. Examples for problems 1.2-1, 1.2-2. Details about exams and dailies. Info about maple tutorials next week. Preview of "The Three Examples" for next time. Info about maple lab 1 due soon, get your print from the web site.

      Week 1 references (documents, slides)
      Three Examples, Fundamental Theorem of Calculus, Method of quadrature, Decay law derivation, Background formulae (6 pages, pdf)
      Three Examples (pdf)
      Three Examples, solved 1.2-1,2,5,8,10 by Tyson Black, Jennifer Lahti, GBG (11 pages, pdf)
      Recipes for 1st, 2nd order and 2x2 Systems of DE PDF Document (2 pages, 50kb)
      Log+exponential Background+Direction fields PDF Document (4 pages, 450k). Decay law derivation. Problem 1.2-2. Direction field examples.
      For more on direction fields, print Direction fields document (typeset, 11 pages, pdf)


      Aug 25: Problems 1.2-2, 1.2-6, 1.6-10 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. The Three Examples were introduced (#1, #2 only). 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.