The Undergraduate Program Committee
Date: May 12, 1999
Lecture | Topics/Sections | Some Standard Examples/Concepts |
---|---|---|
Day 1 | 2.5 - 2.6 | Modeling: decay, mixing, cooling, growth. Physical models and the differential equations which result. |
Day 2 | 2.1, 2.2, 2.3, 2.4 | Review derivation of solution to first order linear, and review separable equations. Mixing problems (equal rate in/out is separable; unequal rate is FOL) |
Day 3 | 5.2 | Series solutions to first order linear or second order constant coefficient. Solve via series, and isolate fundamental solutions as and . Observe and are also solutions. How are and related to and ? Leads to representation of functions by series. |
Day 4 | Define Taylor polynomials; Define geometric series and show when it converges to the rational function ; Discuss the notion of Taylor series, and the notion of an interval of convergence: examples and . | |
Day 5 | Define -series and show when they convrge via an improper integral; state the comparison test and ratio test. Ratio test can be deduced from the comparison test. Use ration test to define the radius of convergence of a power series. | |
Day 6 | Define the notion of absolute/conditional convergence; Discuss alternating series and the error resulting from using partial sums. Consider as an application, the evaluation of (sine, cosine, ). How could one build a function for a calculator via partial sums of the Taylor series for the sine? | |
Day 7 | 3.1 - 3.3 | Review of second order constant coefficient (real roots),
Just use characteristic equation and unmotivated solutions here (proofs in a couple of days) |
Day 8 | 3.4 | Review of complex numbers, complex exponential and second order constant coefficient (complex roots). |
Day 9 | 11.3, 11.4 [Crowell & Slesnick] | Linear Differential Operators (formal ring properties).
Homogeneous and nonhomogeneous solutions
Set of homogeneous solutions forms a vector space General solution of the form |
Day 10 | 11.4 [Crowell & Slesnick] | Solve second-order constant coefficient problem by reducing to a first-order system and using general solution to FOL equations. |
Day 11 | 4.1, 3.3 | Linear independence of solutions. General definition.
Theorem 4.12 [4.1] (w/o Wronskian), that is dimension of space of homog solutions equals the order of the equation. Detecting linear independence: the Wronskian Higher dimensional case after a review of determinants |
Day 12 | 7.2 | Review of matrices, determinants (alternating nature) |
Day 13 | 4.1 | Wronskian (higher order): problem 20 page 207
Take an nth order linear DE and write it as a system. Introduce matrix notation for this motivating Chapter 7. Perhaps revisit Theorem 4.12. |
Day 14 | 3.6 3.7 | [Nonhomogeneous and nonconstant coefficient equations] Finding a particular solution. Undetermined coefficients and variation of parameters. |
Day 15 | 3.8/7.1 | Harmonic Oscillators (simple and not) |
Day 16 | 7.3 | Systems of linear equations, linear independence, eigenvalues and eigenvectors. |
Day 17 | 7.3 | Systems of linear equations, linear independence, eigenvalues and eigenvectors. |
Day 18 | 7.4 | Basic Theory of First order systems |
Day 19 | 7.5 | Homogeneous linear systems with constant coefficients |
Day 20 | 7.6 | Complex Eigenvalues |
Day 21 | 7.7, 10.1 | Repeated eigenvalues, Separation of Variables (heat conduction) |
Day 22 | 10.1 | Heat equation |
Day 23 | 10.2 | Fourier Series |
Day 24 | 10.3, 10.4 | Fourier Convergence Theorem; Even and odd functions |
Day 25 | 10.4, 10.5 | Even and odd functions; More general heat equation |
Day 26 | 10.6 | The Wave Equation |
Day 27 | 10.7 | Laplace's Equation |
Day 28 | Wrap it up | |