MthSc 208, Fall 2010
MthSc 208, Fall 2010
"You see this little hole? This moth's just about to emerge. It's in
there right now, struggling. It's digging it's way through the thick
hide of the cocoon. Now, I could help it - take my knife, gently widen
the opening, and the moth would be free - but it would be too weak to
survive. Struggle is nature's way of strengthening it."
--Locke (Lost, 2004)
Instructor: Dr. Matthew
Macauley
Class: Introduction to Ordinary Differential Equations
Course
Syllabus
Class lecture notes
Section 1:
Introduction to Ordinary Differential Equations. Modeling
physical situations that exhibit exponential growth and exponential
decay. Plotting slope fields using the isocline method. Sketching
slope fields of autonomous differential equations. Approximating
solutions using Euler's method. 9 pages. Last updated January 21,
2011. (Brannan/Boyce: Sections 1.1, 1.3, 2.3, 2.5, & supplemental
material).
Section 2:
First Order Differential Equations. Solving 1st order ODEs using
separation of variables, the integrating factor method, and variation
of parameters. Strucutre of solutions to 1st order linear ODEs, and
connections to parametrized lines. Models of motion with air
resistance. Mixing problems. The logistic equation as a population
model. 21 pages. Last updated February 17, 2011. (Brannan/Boyce:
Sections 2.1, 2.2, 2.3, 2.4, 2.5)
Section 3:
Second Order Differential Equations. Models that use 2nd order
ODEs. Solving homogeneous linear 2nd order ODEs. Solving inhomogeneous
ODEs using the method of undertermined coefficients. Simple harmonic
motion. Harmonic motion with damping and with forcing terms. Solving
2nd order non-constant coefficient ODEs. The power series method, and
the theorem of Frobenius. 29 pages. Last updated February 17,
2011. (Brannan/Boyce: Sections 4.1, 4.2, 4.3, 4.4, 4.7, &
supplemental material).
Section 4:
Systems of Differential Equations. Intro to linear algebra: Adding
and multiplying matrices. Writing systems of linear equations with
matrices, inverses and determinants of 2x2 matrices, eigenvalues and
eigenvectors of 2x2 matrices. Using linear algebra to solve systems
of two 1st order linear ODEs x'=Ax; 3 cases (i) real distinct
eigenvalues, (ii) repeated eigenvalues, (iii) complex eigenvalues. The
SIR model in epidemiology. 26 pages. Last updated October 20,
2010. (Brannan/Boyce: Sections 3.1, 3.2, 3.3, 3.4, 3.5,
A.1)
Section 5:
Laplace Transforms. Definition and properties of the Laplace
transform. Using Laplace transforms to solve ODEs. Using the Heavyside
function to express, and take the Laplace transform of, piecewise
continuous functions. Solving ODEs with discontinuous forcing
terms. Taking the Laplace transform of periodic functions. Impulse
functions and delta functions. 15 pages. Last updated October 20,
2010. (Brannan/Boyce: 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7)
Section 6:
Fourier Series. Introduction to Fourier series -- derivation and
computation. Even and odd functions, and Fourier cosine and sine
series. Complex version of Fourier series. Parseval's identity and
applications to series. 13 pages. Last updated October 29,
2010.
Section 7:
Partial Differential Equations. The (1-dimensional) heat and wave
equations. Analysis of different boundary conditions. Introduction to
PDEs in higher dimensions. Harmonic functions, Laplace's equation, and
steady-state solutions to the heat equation. Solving Laplace's
equation, the heat equation, and the wave equation in two dimensions.
23 pages. Last updated July 29th, 2010.
Lecture summaries
Week 1 summary.
(2 lectures: Section 1, pp. 1-6. Boyce/Brannan Sections 1.1)
Week 2
summary. (4 lectures: Section 1, pp. 7-9. Section 2,
pp. 1-9. Boyce/Brannan Sections 1.3, 2.2, 2.3, 2.4, 2.5)
Week 3
summary (4 lectures: Section 2, pp. 9-19. Boyce/Brannan
Sections 2.1, 2.2, 2.3.)
Week 4
summary. (4 lectures: Section 2, pp. 19-21. Section 3,
pp. 1-9. Boyce/Brannan Sections 2.5, 4.1, 4.2, 4.3, 4.4, 4.6)
Week 5
summary. (3 lectures: Section 3, pp. 10-18. Boyce/Brannan
Sections 4.5, 4.6, 4.7)
Week 6
summary. (4 lectures: Section 4, pp. 1-10. Boyce/Brannan
Sections 3.1)
Week 7
summary. (4 lectures: Section 4, pp. 10-19. Boyce/Brannan
Sections 3.2, 3.3, 3.4)
Week 8
summary. (4 lectures: Section 4, pp. 19-26. Boyce/Brannan
Sections 3.4, 3.5)
Week 9
summary. (4 lectures: Section 5, pp. 1-10. Boyce/Brannan
Sections 5.1, 5.2, 5.3, 5.4, 5.5)
Week 10
summary. (4 lectures: Section 5, pp. 10-15. Section 6,
pp. 1-4. Boyce/Brannan Sections 5.5, 5.6, 5.7, 9.2)
Week 11
summary. (4 lectures: Section 6, pp. 4-12. Boyce/Brannan
Sections 9.2, 9.3, 9.4, supplemental material)
Week 12
summary. (2 lectures: Section 6, pp. 12-13. Section 7,
pp. 1-2. Boyce/Brannan Sections 9.5, supplemental material)
Week 13
summary. (3 lectures: Section 7, pp. 2-7. Boyce/Brannan Sections
9.5, 9.6)
Week 14
summary. (4 lectures: Section 7, pp. 7-15. Boyce/Brannan Sections
9.7, 9.8)
Week 15
summary. (2 lectures: Section 7, pp. 15-17. Boyce/Brannan
Sections 9.8)
Week 16
summary. (3 lectures: Section 7, pp. 18-23. Supplemental
material)
In-class worksheets
Worksheet
1: Separation of variables
Worksheet
2: Integrating factor
Worksheet
3: Mixing problems
Worksheet
4: 2nd order ODEs with constant coefficients
Worksheet
5: Method of undetermined coefficients
Worksheet
6: Basic linear algebra
Worksheet
7: Systems of differential equations (real eigenvalues)
Worksheet
8: Systems of differential equations (complex eigenvalues)
Worksheet
9: Systems of differential equations (repeated eigenvalues)
Worksheet 10: Laplace Transforms
Worksheet 11: Properties of Laplace Transforms
Worksheet
12: Solving ODEs with Laplace Transforms
Worksheet 13: Inverse Laplace Transforms
Worksheet
14: Laplace Transforms and the Heavyside Function
Worksheet
15: ODEs with Piecewise Forcing Terms
Worksheet
16: Fourier Series
Worksheet
17: Complex Fourier Series
Worksheet
18: Parseval's Identity
Worksheet
19: The Heat Equation
Worksheet
20: The Wave Equation
Worksheet
21: The 2D Heat Equation
Homework
Homework 1. Due
Tuesday, August 24th at 4pm.
Homework 2. Due
Friday, August 27th at 4pm.
Homework 3. Due
Monday, August 30st at 4pm.
Homework 4. Due
Friday, September 3rd at 4pm.
Homework 5. Due
Tuesday, September 7th at 4pm.
Homework 6. Due
Friday, September 10th at 4pm.
Homework 7. Due
Tuesday, September 14th at 4pm.
Homework 8. Due
Tuesday, September 21st at 4pm.
Homework 9. Due
Friday, September 25th at 4pm.
Homework 10. Due
Tuesday, September 28th at 4pm.
Homework 11. Due
Friday, October 1st at 4pm.
Homework 12. Due
Friday, October 8th at 4pm.
Homework 13. Due
Friday, October 15th at 4pm.
Homework 14. Due
Tuesday, October 19th at 4pm.
Homework 15. Due
Friday, October 22nd at 4pm.
Homework 16. Due
Friday, October 29th at 4pm.
Homework 17. Due
Friday, November 5th at 4pm.
Homework 18. Due
Friday, November 12th at 4pm.
Homework 19. Due
Wednesday, November 17th at 4pm.
Homework 20. Due
Tuesday, November 23rd at 4pm.
Homework 21. Due
Friday, December 3rd at 4pm.