PROPERTIES OF LINEAR SYSTEMS AND THE LINEARITY PRINCIPLE Section 3.1 PROPERTIES OF LINEAR SYSTEMS AND THE LINEARITY PRINCIPLE

Intro to linear systems Lots of useful systems of DEs are in the form Examples include the damped (or undamped) harmonic oscillator (see p. 234) and the competing stores model (see p. 235). Use LinearPhasePortraits to experiment with different values of a, b, c, and d.

Equilibrium solutions Find the equilibrium solution(s) to Do the values of a, b, c, and d have an effect on the equilibrium solution(s)?

Competing stores model Read p. 235-237. Quantities: x(t) = daily profit of Paul’s store at time t. y(t) = daily profit of Bob’s store at time t. a = effect of Paul’s profits on the change in Paul’s profits. Explain: a > 0 means ___; a < 0 means ___. b = effect of Bob’s profits on the change in Paul’s profits. Explain: b > 0 means ___; b < 0 means ___. Explain c and d. Do p. 252 1-4.