MthSc 208, Summer Session II, 2012
MthSc 208, Summer Session II, 2012
"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."
Class: Introduction to Ordinary Differential Equations.
Instructors: Dr. Tim
Teitloff (June 27--July 16), and
Dr. Matt Macauley (July
Class lecture notes
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
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)
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, &
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,
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)
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)
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)
1a: Plotting slope fields
2a: Separation of variables
2b: Integrating factor
2c: Mixing problems
3a: 2nd order ODEs with constant coefficients
3b: Method of undetermined coefficients
3c: Mass-spring systems
4a: Basic linear algebra
4b: Systems of differential equations (real eigenvalues)
4c: Systems of differential equations (complex eigenvalues)
4d: Systems of differential equations (repeated eigenvalues)
Worksheet 5a: Laplace Transforms
Worksheet 5b: Properties of Laplace Transforms
5c: Solving ODEs with Laplace Transforms
Worksheet 5d: Inverse Laplace Transforms
5e: Laplace Transforms and the Heavyside Function
5f: ODEs with Piecewise Forcing Terms
6a: Fourier Series
6b: Complex Fourier Series
6c: Parseval's Identity
7a: The Heat Equation
7b: The Wave Equation
7c: The 2D Heat Equation
Homework 1. Due
Friday, June 29th.
Homework 2. Due
Tuesday, July 3rd.
Homework 3. Due
Friday, July 6th.
Homework 4. Due
Tuesday, July 10th.
Homework 5. Due
Monday, July 16th.
Homework 6. Due
Friday, July 20th.
Homework 7. Due
Tuesday, July 24th.
Homework 8. Due
Friday, July 27th.
Homework 9. Due
Tuesday, July 31st.
Homework 10. Due
Friday, August 3rd.