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.