While the lecture notes contain an outline of the material covered in lecture, other material is often written on them during lecture. (All of the jokes are left out.)
Topics include basic course business, a quick introduction to some ODE terminology, and a couple of worked examples.
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.
Discuss two more models (logistic growth and heat loss) with a little graphical analysis of the logistic growth model. Start into the basic theory for linear differential equations: the general solution can be built in stages.
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.
A little work with the Fundamental Theorem of Calculus and the method of Variation of Parameters, followed by an introduction to numerical methods. (Practice pronouncing Euler and Heun...)
Begin with an example of Heun's method (the Modified Euler Method) followed by
two new ways to solve linear equations when you have constant
coefficients. The characteristic equation method gives an easy
route to the homogeneous solution. The method of undetermined
coefficients gives a fairly simple way to construct particular
solutions when the forcing term is nice.
Contents: more Undetermined Coefficients, a little bit of qualitative analysis (graph the solution without finding the solution formula) and the beginning of the review for Test #1.
Review Day; visit the Exams and Review Sheets page.
Topics included stability for steady state solutions, studying stability by graphical analysis as well as linearization about the steady state. Free and forced solutions for linear equations were introduced. Springs and masses will arrive on Monday.
Topics included the derivation of the basic spring-mass model along with some qualitative analysis of the model. The model for the simple pendulum was also discussed.
Topics included the predator-prey model, with some graphs in the phase plane, and the first steps in solving second-order, linear differential equations. The Wronskian and linear independence were, perhaps, the really new ideas here.
Today was the day for characteristic equations. The method gives you the general solution to a homogeneous, constant-coefficient differential equation. There is a little bit of extra work if the characteristic equation has complex roots or a single real root.
Today was the day for 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 also returned for second-order equations.
The lecture focused on questions about the qualitative behavior for spring-mass systems. When is the damping strong enough to prevent oscillations? How does the amplitude of the solution depend on the initial data? How does the amplitude of the solution depend on the frequency of the forcing function?
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.
The lecture focused on analysis of solutions in the phase plane.
The first examples were unforced springs with various damping
coefficients. The solutions "spiral" into the origin as long as
there is no forcing function. When periodic forcing is added, the
solutions converge to a (stable) limiting periodic solution. The
second example studied was 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
There is also a
Maple Worksheet with some of
the examples used in the lecture.
And, if that wasn't enough, here are post-script versions of the
Scanning for Resonance
The lecture focused on linear systems with constant coefficients. The characteristic equation provides the form of the exponentials in the solution. There is a little more work to build linearly independent solution pairs for the general solution.
The lecture focused on linear systems again. There was one
example with eigenvalues and one example with forcing terms.
For some simple cases, you can use undetermined coefficients to
find a particular solution.
Here is the Maple Worksheet used in the lecture.