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 linear systems. More next time...
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 with forcing terms. The characteristic equation provides the homogeneous solution. For some simple cases, you can use undetermined coefficients to find a particular solution.
REVIEW DAY!