Topics include basic course business, a quick introduction to some ODE terminology, and a couple of worked examples. Here is a copy of the Maple Worksheet used during the lecture.
Two ways to build an ODE model: one way focuses on data and fitting curves to data. The second (more general) view combines a balance equation with the information obtained from the data. Here is a copy of the population data worksheet used during the lecture.
Discuss two more models (heat loss and epidemics). Start into the basic theory for linear differential equations: the general solution can be built in stages. What is a homogeneous solution? What is a particular solution?
More work on the basic theory of linear equations. The homogeneous equation is always separable and so you can solve it by integration. The particular solution is a little harder to find, but you can use the homogenous solution to build it. The method is called variation of parameters.
Here is a copy of the Maple Worksheet used during the lecture.
A little review of the course so far, followed by a visit with the Fundamental Theorem of Calculus. Two new methods for analyzing linear equations when the coefficients are constant: the method of Characteristic Equations give the homogeneous solution and the method of Undetermined Coefficients builds a particular solution.
More about solving constant-coefficient equations. Go back to
graphing solution curves without solution formulas.
Contents: beginning of the review for Test #1, with a little numerical analysis on the side.
Review Day; visit the Exams and Review Sheets page.
Topics include a little bit of general theory, the introduction of linear independence and building the homogeneous solution, and a start on the characteristic equation method.
Finish off the homogeneous solution with the complex conjugate case. Bring in a forcing term and recall Undetermined Coefficients. The method builds the particular solution for a non-homogeneous, constant-coefficient differential equation. There is a little bit of extra work if the forcing function happens to solve the corresponding homogeneous equation (but you knew that already). Euler's method returns for second-order, encore performance.
The lecture focused on questions about the qualitative behavior for spring-mass systems (again!). The starting point was the graph for a solution of a damped, forced system to illustrate the existence of limiting periodic solutions. The details behind this included the "proof" that the homogeneous solution for a damped system is always transient. The final topic was a really quick introduction to graphing solution in the phase plane.
Here is a copy of the Maple Worksheet used during the lecture.
The lecture startw with analysis of solutions in the phase plane. The new example is a model for competing species. The system has a steady state in which the species co-exist. The stability of the system requires some new tools for analyzing linear systems. There is also a Maple Worksheet with some of the examples used in the lecture.
The lecture focuses on linear systems again. with some curve sketching at the start. The complex conjugate case returns, with a non-eigenvector approach. The review begins!
There is also a Maple Worksheet with some of the examples used in the lecture.
REVIEW DAY!