﻿

MATH 310-02: ORDINARY DIFFERENTIAL EQUATIONS (SPRING 2012)

COURSE INFORMATION

 Textbook Differential Equations and Boundary Value Problems: Computing and Modeling (C. Henry Edwards & David E. Penney, 4th Edition, Prentice Hall) Time 11:30am-12: 20pm, MWF Location NICCOL 006 Instructor Phone 208-885-6629 Office 326 Brink Hall Office hours MWF: 2-3 PM Webpage:

Topics

We will cover (more or less) the following topics

1)    First-Order Differential Equations (Chapter 1)

2)    Mathematical Models and Numerical Methods (Chapter 2)

3)    Linear Equations of Higher Order (Chapter 3)

4)    Introduction to Systems of Differential Equations (Chapter 4)

5)    Linear Systems of Differential Equations (Chapter 5)

6)    Nonlinear Systems and Phenomena (Chapter 6)

7)    Laplace Transform Methods (Chapter 7)

8)    Power Series Methods (Chapter 8)

Exams

1)    Midterm exam 1: Friday, Feb 10, 2012. Time: 11:30 am-12: 20 pm. Location: NICCOL 006.

2)     Midterm exam 2: Friday, March 30, 2012. Time: 11:30 am-12: 20 pm. Location: NICCOL 006.

3)    Final exam: 10 AM – 12PM. Tuesday, May 8. Location: NICCOL 006.

Homework & Quiz Policy

1)    Homework: For each set, some random problems will be graded.

2)    Suggested problems: Not to be graded, just for practice.

3)    Quizzes: Take place randomly. The questions are normally from the lectures.

1)    Homework & Quizzes: 15%

2)    Exam 1: 20%

3)    Exam 2: 30%

4)    Final: 35%

1)    No late homework.

2)    Grades are only discussed in office hours or by appointment. No grade review in or around class time.

Warning: The lecture notes are posted for the students’ convenience. However, they might not contain everything covered in the class. Moreover, they might have some minor errors/typos. The students are supposed to attend the classes and make the changes in the lecture notes accordingly.

1) Lecture 0 (Jan 11, 2012): Introduction to the course.

Content: Introduction to Differential Equations, Basic Notations and Definitions, some other information about the course.

Note: This lecture does not follow the textbook. The purpose is to give an overview of the course. From the next lecture, we will follow the textbook.

2) Lecture 1 (Jan 13, 2012): First Order Differential Equations.

Content: General Form, Physical Models related to 1st-Order ODEs, Initial Value Problem.

Note: This lecture + previous one is approximately section 1.1.

3) Lecture 2 (Jan 18, 2012): Integral as General and Particular Solutions.

Content: First Order ODE, Second ODEs, Velocity and Acceleration.

4) Lecture 3 (Jan 20, 2012): Slope Fields and Solution Curves

Content: Slope, Derivative as Slope, Solution Curve, Using Slope Fields to Approximate Solution Curves.

5) Lecture 4 (Jan 23, 2012): Separable Equations and Applications

Content: Separable Equations, Natural Growth and Decays, Heating and Cooling.

6) Lecture 5 (Jan 25, 2012): Linear First-Order Differential Equations

Content: Integrating Factors, Mixture Problem.

7) Lecture 6 (Jan 27, 2012): Substitution Methods

Content: Substitution Method, Homogeneous Equations.

8) Lecture 7 (Jan 30, 2012): Exact Equation

Content: Partial Derivative, Exact Equation, Implicit Solutions.

9) Lecture 8 (Feb 1, 2012): Population Models

Content: Logistic Equation, Limiting Population, Carrying Capacity.

10) Lecture 9 (Feb 3, 2012): Equilibrium Solutions and Stability

Content: Autonomous Equation, Critical Point, Stable and Unstable, Phase Diagram, Logistic Equation with Harvesting.

11) Lecture 10 (Feb 6, 2012): Numerical Approximation: Euler’s Method

Content: Slope, Derivative as Slope, Euler Method.

Note: More files are posted in Blackboard. Check them out!

12) Review for Exam 1: (Feb 8, 2012):

Content: Basic things to keep in mind.

13) Lecture 11 (Feb 13, 2012): Second-Order Linear Equations

Content: Second-Order Linear Equation, Superposition Principle, Linear Dependence, Wronskian, General Solution.

14) Lecture 12 (Feb 15, 2012): Second-Order Linear Equations (cont)

Content: Constant Coefficient Equation, Characteristic Equation, Distinct Roots, Repeated Root.

15) Lecture 13 (Feb 17, 2012): General Solutions of Linear Equations

Content: nth-order Linear Equation, Homogeneous Equation, Superposition Principle, Linear Independence, General Solution, Wronskian.

16) Lecture 14 (Feb 22, 2012): Homogeneous Equations with Constant Coefficients

Content: Characteristic Equation, Distinct Roots, Repeated Roots.

17) Lecture 15 (Feb 24, 2012): Homogeneous Equations with Constant Coefficients (Cont)

Content: Complex Roots.

18) Lecture 16 (Feb 27, 2012): Nonhomogeneous Equations and Undetermined Coefficients

Content: Complementary Function, Particular Solution, Method of Undetermined Coefficients.

19) Lecture 17 (Feb 29, 2012): Nonhomogeneous Equations and Undetermined Coefficients

Content: Exceptional Cases.

20) Lecture 18 (March 2nd, 2012): Nonhomogeneous Equations and Variation of Parameters

Content: Variation of Parameter Method.

21) Lecture 19 (March 5th, 2012): Forced Oscillation and Resonance

Content: Review for Second-Order Linear Equations with Constant Coefficients, Application to Forced Oscillation, Resonance.

22) Lecture 20 (March 7th, 2012): Forced Oscillation and Resonance

Content: Continuation of Lecture 19.

Note: The (current) new file was posted on March 7th at 3:00 PM. If you downloaded the file before that, please redo it.

23) Lecture 21 (March 9th, 2012): First-Order Systems and Applications

Content: Mixture Problem with Two Tanks, First-Order System v.s. Higher Order Equation, Solving Linear System, Definition of Linear System.

24) Lecture 22 (March 19th, 2012): Linear First-Order Systems and Method of Elimination

Content: Definition of Linear System, Examples, Method of Elimination.

25) Lecture 23 (March 21st, 2012): Matrices and Linear Systems

Content: Matrix, Vector, Basic Operations, Vector Function, System as Equation of Vector Function, General Solution.

Note: a typo corrected at 3:00 PM, Friday, March 23rd.

26) Lecture 24 (March 23rd, 2012): Matrices and Linear Systems (Part II)

Content: General Solutions of Linear Systems.

Note: a typo corrected at 3:00 PM, Friday, March 23rd.

27) Lecture 25 (March 26st, 2012): Eigenvalue Method for Homogeneous System

Content: Method to Find Eigenvalues and Eigenvectors, Solution of Linear System.

Note: posted in advance on Friday, March 23rd.

Practice problems:

1) Solve the problems 1, 3, 4 in section 4.2 by two different methods: Method of Elimination and Eigenvalue Method.

2) Solve the problem 2 in Section 4.2 by the method of elimination.

28) Review for Exam 2: (March 28th, 2012):

Content: Topics and Practice Problems.

29) Lecture 26 (April 2nd, 2012): Laplace Transform

Content: Improper Integral, Laplace Transform

30) Lecture 27 (April 4th, 2012): Laplace Transform and Inverse Transform

Content: Laplace Transform, Inverse Laplace Transform, Piecewise Continuous Function, Unit Step Function.

31) Lecture 28 (April 6th, 2012): Laplace Transform and Inverse Transform & Transformation of Initial Value Problems

Content: Piecewise Continuous Function, Unit Step Function, Laplace Transform of Derivatives, Transformation of Initial Value Problems.

32) Lecture 29 (April 9th, 2012): Transformation of Initial Value Problems

Content: Solve the Initial Value Problems by the Laplace Transform.

33) Lecture 30 (April 11st, 2012): Product of Transforms

Content: Convolution, Laplace Transform of Convolution, Inverse Laplace Transform of Product.

34) Lecture 31 (April 13st, 2012): Piecewise Continuous Input Functions

Content: Translation, Laplace Transform of Translation and Its Applications.

35) Lecture 32 (April 16th, 2012): Laplace Transform: Problems and Applications

Content: Solving Second-Order Linear Equation by the Laplace Transform.

NOTE: File updated, Graphs corrected (at 3 PM, Monday April 16th).

36) Lecture 33 (April 18th, 2012): Power Series

Content: Definition of Power Series, Convergence, Radius of Convergence, Power Series of Derivative.

37) Lecture 34 (April 20th, 2012): Power Series (Cont)

Content: Power Series of Derivative, Solving Differential Equation by the Series Method.

38) Lecture 35 (April 23th, 2012): Power Series (Cont)

Content: Solving Differential Equation by the Series Method, Radius of Convergence, Initial Value Problem.

39) Lecture 36 (April 25th, 2012): Series Solutions Near Ordinary Points

Content: Analytic Functions, Ordinary Point, Singular Point, Solutions Near Ordinary Points. Guaranteed Radius of Convergence.

40) Lecture 37 (April 27th, 2012): Series Solutions Near Ordinary Points (Cont)

41) Lecture 38 (April 30th, 2012): Series Solutions Near Ordinary Points (Cont)

Content: Two-term recurrence relation v.s Three-term recurrence relation.

42) Final Review 1 (May 2nd, 2012): Laplace Transform Method

Content: Topics and practice problems.

43) Final Review 2 (May 4th, 2012): Power Series Method

Content: Topics and practice problems.

44) Final Extra Review (May 7th, 2012):

HOMEWORK

1)     Homework 1 (Due Friday, Jan 27, 2012):

For submission:

Section 1.1: 6, 7, 9, 19, 20

Section 1.2: 1, 6, 7, 8, 27, 28

Section 1.3: 1, 2, 3, 14, 15

Suggested:

Section 1.1: 10, 11, 12, 25, 26

Section 1.2: 9, 10, 40.

2)     Homework 2 (Due Friday, Feb 3rd, 2012)

For submission:

Section 1.4: 1, 2, 3, 4, 9, 14, 19, 20, 21, 43.

Section 1.5: 1, 3, 5, 7, 19, 36, 37.

Section 1.6: 1, 2, 3, 4, 17.

Suggested:

Section 1.4: 17, 18.

Section 1.5: 20-25.

3)     Homework 3 (Due Monday, Feb 13rd, 2012)

For submission:

Section 1.6: 31, 33, 35, 36.

Section 2.1: 1, 2, 5, 6, 21, 29.

Section 2.2: 1, 2, 4, 5, 13, 14.

Suggested:

Section 2.2: 21, 22, 23, 24.

4)     Homework 4 (Due Friday, Feb 24, 2012)

For submission:

Section 2.4: 11-13. Note: for the Euler’s method, you might use Excel as demonstrated in the class.

Section 3.1: 2, 3, 33, 34, 35, 39, 40.

Suggested:

Section 3.1: 52, 53.

5)     Homework 5 (Due Monday, March 5, 2012)

For submission:

Section 3.2: 1, 3, 7, 8, 13, 14.

Section 3.3: 13, 15, 21, 22, 23.

Section 3.5: 1, 2, 3, 4.

Suggested:

Section 3.3: 20, 39, 40.

Section 3.5: 5.

7)     Homework 7 (Due Friday, April 13 2012)

Section 7.1:

Compute the Laplace Transform Using Definition: 1-3. Suggested: 4-6.

Compute the Laplace Transform (using any method you like): 7-10.

Compute the Laplace Transform (using the table of Laplace Transform): 13, 15, 16, 19. Suggested: 17, 18, 20, 21, 22.

Compute the Inverse Laplace Transform: 23, 26, 28, 30. Suggested: 32.

8)     Homework 8 (Due Friday, April 20 2012)

Section 7.2:  3-6, 10. Suggested: 11-16.

Section 7.4: 1-3, 7-10. Suggested: 4-6, 11-14.

Section 7.5: 1, 2, 4, 6, 33, 34. Suggested: 7-10.

9)     Homework 9 (Due Friday, April 27 2012)

Section 8.1:  1-6, 11-14, 19-20. Suggested: 15-18.

10)   Homework 10 (Due Friday, May 4 2012)

Section 8.2:  1-4, 16, 17 ,18, 19, 23, 24.