[Goeffrey Hunter] Wed Tutorial:
Only 3 students showed on Wed (rainy day).
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.
Students asked how they can use Maple to solve Euler's method. I
suggested an alternate solution method, shown below.
[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 := y0:
> X := 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
Heaviside coverup method manuscript
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 Tuesday: 3.1 problems.
Weeks 5,6 slides
Linear equations, reduced echelon, three rules (typeset, 7 pages, 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 12,13,14: 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=20, using units of millions of cubic feet. You should have
obtained the model x'=1/4 -x/16, x(0)=20.
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.
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)
13 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 14 Sep, Page 86, 2.1: 8, 16
Due 14 Sep, Maple Lab 1: 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.
Discussed 2.2 problems, stability theory, phase diagrams,
calculus tools, DE tools, partial fraction methods.
PDF Document (4 pages, 86k)
Introduction to RECT, TRAP and SIMP rules. 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.
Some 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 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 5,6,7: Sections 2.1,2.2,2.3,2.4.
04 Sep: Holiday
05 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.
06 Sep: Begin maple 1 in Wednesday lab, JFB 102.
[Geoffrey Hunter]: Started maple lab 1. 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.
07 Sep: Collected in class, Page 41, 1.4: 18, 22, 26.
Collected 12 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 29,30,31: Sections 1.3,1.4,1.5.
29 Aug: Collected in class Page 16, 1.2: 2, 4, 6, 10.
Problems 1.2 discussed in class. Maple and Matlab methods. Integral
table methods. The Three Examples introduced in week 1 were continued.
Panels 1 and 2 in the answer check for an initial value problem like
1.2-2: y'=(x-2)^2, y(2)=1. 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,
which is Theorem 1 in Edwards-Penney, 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)
Peano and Picard Theory (3 pages, pdf)
Picard-Lindelof and Peano Existence Example, similar to 1.3-14 (1 page, pdf)
30 Aug, More examples of how to write reports, 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.
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.
31 Aug: Collected in class, Page 26, 1.3: 14. The dailies on Page
41, 1.4: 6, 12 will be due Tuesday 5 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.
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)
Week 1, Aug 24: Sections 1.1,1.2.
Aug 24: Sections 1.1, 1.2. Examples for problems 1.2-1, 1.2-2,
1.2-6, 1.2-10. Details about exams and dailies. Maple tutorial next
week, part of the lectures. Preview of "The Three Examples" for next
time. Info about maple lab 1, which is due soon; get your print from the web site.
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.
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.
Week 1 references (documents, slides)
Three Examples, Fundamental Theorem of Calculus, Method of quadrature, Decay law derivation, Background formulae (6 pages, 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)