UBC Math 215/255 Fall 2009

Section 102 at 9am

This page is for students of section 102 with instructor Tsai only. In particular, students from other sections please do not come to the following office hours.

Information

Lecture Time & Location:

MWF 9:00-9:50am, LSK 200.

Instructor:

Dr. Tai-Peng Tsai, Math building room 109, phone 604-822-2591, ttsai@math.ubc.ca.

Office hours for Section 102:

Tue 2pm, Wed 1pm, Thu 4pm, and by appointment (Tsai's schedule).

• Problem sets and Exams

Lecture summary and Announcements

L1 09.09

• Course outline.
• Examples and terminology (S1.3).

L2 09.11

• Direction field (S1.1).
  java scripts for drawing direction fields and phase planes: dfield and pplane
• S2.1 Linear first order equations and integrating factor
• S2.1 Ex 1: Solve y'+cos t y = 0.
• S2.1 Ex 2: Solve y' + y = e^t.

L3 09.14

• S2.1 Ex 3: Solve y'-2ty = t, y(0)=1.
• S2.2 Separable equations.
• S2.2 Ex 1: Solve dy/dx = 4x(1-y), y(0)=1+e.

L4 09.16

• S2.2 Ex 2: Solve dy/dx = 4x/(1-y), and draw the graphs of several integral curves. Also find the solution passing through (3,0) and determine its interval of validity.
• S2.2 Ex 3: Repeat Ex 2 for dy/dx = (x-x^3)/(y^3-y/2) and the point (1,1).
  Graph of several integral curves: [jpg]. Interval of validity: 1 /√ 2 < x < √ 3/ √ 2
  (The figure is generated by a MatLab code. Other common choices include Mathematica and Maple.)

L5 09.18

• S2.3 Modeling
• Ex 1. Water tank problem.
• Ex 2. Interest problem.
• Ex 3. Falling body with air resistence.

L6 09.21

• Ex 3 (continue)
  To find the root of f(t)=0 in MatLab, run:
  f=@(t)318.45-44.1*t-288.45*exp(-t/4.5); x=fzero(f,[5,6])
• S2.5 Population dynamics: logistic model and stability.

L7 09.23

• Ex 1 (application of logistic model: max number of UBC students)
• Ex 2 (stability of const solutions by graph of f(y))
• S2.6 Exact equations: motivation, definition and equivalence condition M_y = N_x.

L8 09.25

• Ex 1 y+2x + (x-3y^2)dy/dx=0.
• Ex 2 sin y + (1+x)cos y dy/dx=0.
• Integrating factor, Ex 3 (-x^2-e^y+2x) + e^y dy/dx=0.

L9 09.28

• Ex 4 Find mu(y) for 2x + (x^2+2e^y)dy/dx=0.
• S2.4 Unique existence theorem.
• S3.1 2nd order linear equation with constant coefficients.
• Ex 1 y'' - 4y=0.

L10 09.30

• characteristic eqaution.
• Different real roots case. Ex 2 2y''-5y'-3y=0.
• S3.2 Basic properties.
• Unique existence theorem.

L11 10.02

• Midterm on 10.09: up to and include S3.2, 50 minutes, 2 Yes/No problems with reasons needed, 3 computation problems, no notes and no calculators.
• Principle of superposition.
• Differential operator.
• Fundamental set and Wronskian.

L12 10.05

• S3.3 complex roots case.
• Complex numbers. Ex 1: Evaluate exp(2+pi i/4)
• Taylor series and Euler's formula.
• Ex 2. Find the general solution of y''-2y'+10y=0. (b) Find the solution satisfying y(0)=2, y'(0)=-1.

L13 10.07

• Sample midterm exam: MT1 of Fall 2007 and its solutions. Note we have Yes/No questions.
• Tutorial service: In additional to those provided by Math Department and AMS, a new TA will be tutoring for Math 255, 256, and 257 in front of Starbucks in the Kaiser building (on Main Mall), on Monday-Wednesday-Friday from 3-5 pm. The TA is a PhD student in Mechanical Engineering.
• Ex 3. Solve 9y''+6y'+37y=0, y(0)=3, y'(0)=0.
• S3.4 repeated roots case.
• Ex 1. Solve y''-6y'+9y=0, y(0)=2, y'(0)=3.
• Midterm survey.

Midterm Exam I 10.09

• Solutions to Midterm Exam I.

L14 10.14

• The final exam is scheduled on Wednesday, December 16, 12pm. The location will be announced later.
• Ex 2. The equation t62y''+2ty'-2y=0 (t>0) has a solution y_1(t)=t. Use the method of reduction of order to find a second solution.
• S3.5. Nonhomogeneous equations and the method of undetermined coefficients.
• Ex 1. Find a particular solution of y''+2y'+2y=2t^2-2.

L15 10.16

• Remarks on midterm exam 1 and the survey.
• Ex 2 -- Ex 7.

L16 10.19

• More examples and summary.
• S3.6 Method of variation of parameters.
• Idea and examples.

L17 10.21

• The final exam is scheduled on Wednesday, December 16, 12pm. You should inform me ASAP if there is an Exam Comflict.
• The TA in the Kaiser building is dedicated to Math 215/255. Thus you won't need to explain to an unspecialized TA.
• The second UBC/UMC (Undergraduate Mathematics Colloquium) talk will be held this WEDNESDAY, OCTOBER 21 from 3:00 to 4:00 in GEOG 214. Richard Anstee will be giving a talk entitled "If you can't square the circle, then at least you can square the square". Interested students are encouraged to attend. An abstract and other information can be found at www.math.ubc.ca/~fsl/UMC.html .
• S3.7 Mechanical and electrical vibration.
• Spring-mass system, undamped forcefree case.

L18 10.23

• Spring-mass system, damped forcefree case.
• Electric circuit system.
• S6.1 Laplace transform.
• Improper integral and piecewise continuous functions.

L19 10.26

• Laplace transform: definition and examples.
• S6.2 Solving DE with Laplace transform.
• Laplace transform of derivatives.

L20 10.28

• S6.3 Step functions. Heaviside functions.
• Translation in t formula.

L21 10.30

• Translation in s formula.
• S6.4 DE with discontinuous forces.

L22 11.02

• Table of Laplace transform, to be provided in the second midterm exam.
• S6.5 Dirac delta function.
• DE with impulse functions as the force.
• S6.6 Convolution.

L23 11.04

• Examples of convolution.
• Chapter 7: linear systems of 1st order DE

L24 11.06

• Second Midterm Exam on 11.13: It will cover sections 3.3-3.7 and 6.1-6.6. 50 minutes. 4 computation problems. No notes and no calculators. A Table of Laplace transform will be provided. Sample midterm exam and its solutions -- Problem 3 belongs to Chapter 7 and is not covered. There will be a TA review session on Tuesday at 11am, location TBA.
• The solution set of homogeneous system of 1st order DE is a vector space.
• S7.5 Homogeneous linear systems with constant coefficients
• Different real roots case: Example 1.

L25 11.09

• Midterm Exam TA Review Session on Tuesday Nov 09 at MATH 204 will review the following problems: p1, p2, p3, p4. (Disregard the third problem in p3 since it is from section 3.8)
• Office hours on Thursday Nov 12: 11AM-noon and 4PM-5:30PM.
• Another three examples.
• S7.6 complex roots case: Example 1

11.11

• The final exam will be held on Dec 16 at 12:00pm. The location for section 102 (taught by Tsai) is MATH 100. See complete list for other sections and courses.
• Partial solutions for the TA review session.
• Office hours on Thursday Nov 12 are revised to: 10AM-11:50AM, 2PM-3PM and 4PM-5:30PM.

Midterm Exam II 11.13

• Solutions to Midterm Exam II.
• Average: 25.06 out of 40, or 65.15%.

L26 11.16

• Finished S7.6: complex eigenvalue case for linear systems.
• S7.7: Fundamental matrix for linear systems.

L27 11.18

• Finished S7.7.
• S7.8: Repeated eigenvalue case.

L28 11.20

• Finished S7.8.
• S7.9: Nonhomogeneous linear systems.

L29 11.23

• Finished S7.9.

L30 11.25

• S9.1.
• S9.2. Autonomous systems, critical points, stability.

L31 11.27

• Finished S9.2.
• Started S9.3: Linear approximation near a critical point.

L32 11.30

• Example using Jacobian matrix.
• S9.4 Competing species.

L33 12.02

• Examples of coexistence and non-coexistence.
• S9.5 Predator-Prey models.

L34 12.04

• Review for final exam.
• See announcements for contents of final exam.
• Office hours during exam period: Dec 8, 11, 14, 15, 2pm-3pm.