MthSc 208, Fall 2011

# MthSc 208, Fall 2011

"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

### 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 December 9, 2011. (Brannan/Boyce: 9.2, 9.4)

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. (Brannan/Boyce: 9.5, 9.6, 9.7, 9.8)

### Lecture summaries

Week 1 summary. (3 lectures: Section 1, pp. 1-6. Boyce/Brannan Sections 1.1, 2.2, 2.4)
Week 2 summary. (4 lectures: Section 1, pp. 7-9, Section 2, pp. 1-9. Boyce/Brannan Sections 1.2, 1.4, 2.1, 2.2, 2.3)
Week 3 summary. (4 lectures: Section 2, pp. 9-16. Boyce/Brannan Sections 1.2, 2.2, 2.3)
Week 4 summary. (4 lectures: Section 2, pp. 16-21, Section 3, pp. 1-3. Boyce/Brannan Sections 2.2, 2.4, 4.1, 4.2, 4.3)
Week 5 summary. (3 lectures: Section 3, pp. 4-10. Boyce/Brannan Sections 4.1, 4.2, 4.3)
Week 6 summary. (4 lectures: Section 3, pp. 10-18, Section 4, pp. 1-2. Boyce/Brannan Sections 3.1, 4.4, 4.5, 4.6)
Week 7 summary. (4 lectures: Section 4, pp. 2-14. Boyce/Brannan Sections 3.1, 3.2, 3.3)
Week 8 summary. (4 lectures: Section 4, pp. 14-22. Boyce/Brannan Sections 3.3, 3.4)
Week 9 summary. (3 lectures: Section 4, pp. 22-26. Boyce/Brannan Sections 3.5, 7.P.1)
Week 10 summary. (3.5 lectures: Section 5, pp. 1-8. Boyce/Brannan Section s 5.1, 5.2, 5.3, 5.4)
Week 11 summary. (3 lectures: Section 5, pp. 8-13. Boyce/Brannan Sections 5.5, 5.6)
Week 12 summary. (4 lectures: Section 5, pp. 13-15, Section 6, pp. 1-7. Boyce/Brannan Sections 5.6, 5.7, 9.2)
Week 13 summary. (4 lectures: Section 6, pp. 8-13, Section 7, pp. 1-2. Boyce/Brannan Sections 9.5)
Week 14 & 15 summary. (5 lectures: Section 7, pp. 2-12. Boyce/Brannan Sections 9.5, 9.6, 9.7)
Week 16 summary. (4 lectures: Section 7, pp. 13-23. Boyce/Brannan Sections 9.8)

### In-class worksheets

Worksheet 1a: Plotting slope fields
Worksheet 2a: Separation of variables
Worksheet 2b: Integrating factor
Worksheet 2c: Mixing problems
Worksheet 3a: 2nd order ODEs with constant coefficients
Worksheet 3b: Method of undetermined coefficients
Worksheet 3c: Mass-spring systems
Worksheet 4a: Basic linear algebra
Worksheet 4b: Systems of differential equations (real eigenvalues)
Worksheet 4c: Systems of differential equations (complex eigenvalues)
Worksheet 4d: Systems of differential equations (repeated eigenvalues)
Worksheet 5a: Laplace Transforms
Worksheet 5b: Properties of Laplace Transforms
Worksheet 5c: Solving ODEs with Laplace Transforms
Worksheet 5d: Inverse Laplace Transforms
Worksheet 5e: Laplace Transforms and the Heavyside Function
Worksheet 5f: ODEs with Piecewise Forcing Terms
Worksheet 6a: Fourier Series
Worksheet 6b: Complex Fourier Series
Worksheet 6c: Parseval's Identity
Worksheet 7a: The Heat Equation
Worksheet 7b: The Wave Equation
Worksheet 7c: The 2D Heat Equation

### Homework

Homework 1. Due Monday, August 29th at 4pm.
Homework 2. Due Friday, September 2nd at 4pm.
Homework 3. Due Monday, September 5th at 4pm.
Homework 4. Due Monday, September 12th at 4pm.
Homework 5. Due Friday, September 16th at 4pm.
Homework 6. Due Wednesday, Septemmber 21st at 4pm.
Homework 7. Due Monday, September 26th at 4pm.
Homework 8. Due Friday, September 30th at 4pm.
Homework 9. Due Wednesday, October 5th at 4pm.
Homework 10. Due Monday, October 10th at 4pm.
Homework 11. Due Friday, October 14th at 4pm.
Homework 12. Due Monday, October 24th at 4pm.
Homework 13. Due Friday, October 28th at 4pm.
Homework 14. Due Thursday, November 3rd at 4pm.
Homework 15. Due Friday, November 11th at 4pm.
Homework 16. Due Wednesday, November 16th at 4pm.
Homework 17. Due Monday, November 21th at 4pm.
Homework 18. Due Tuesday, November 29th at 4pm.
Homework 19. Due Friday, December 2nd at 4pm.
Homework 20. Due Thursday, December 8th at 4pm.
Homework 21. Due Thursday, December 15th at 4pm.