"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 (

Course Syllabus

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.

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.

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.

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.

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.

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.

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)

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 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.