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Explore properties of linear systems and the linearity principle, with examples like harmonic oscillators and the competing stores model. Experiment with equilibrium solutions and values to analyze their effects. Quantities and explanations included. Dive deeper into the topic with readings and exercises provided.
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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.