Question

Assignment #3 Modeling and the Geometry of Systems 1. For this problem, we study the nonlinear differential equation: dx dt = y dy dt = x − x^3 − y^3 a) Algebraically determine all of the equilibria to the differential equation . b) For a solution {x(t), y(t)} with {x(0), y(0)} = {x0, y0}, use your phase diagram to describe the long term behavior of the solution. 1. {x0, y0} = {1, 1} 2. {x0, y0} = {−1, −1} 3. {x0, y0} = {2, −1.5} 4. {x0, y0} = {1.5, −2} 2. A slinky is dangled from atop a high building, and its relative position is modeled by the second order differential equation y" + 2y' + y = 0 where y is the height of the bottom of the slinky. a) Convert this second order differential equation into a 2-dimensional first order differential equation using the variable v = y 0 . b) Write an interpretation of your graphs in terms of the typical behavior of a slinky. c) Sketch by hand both the y(t) and v(t) graphs for the initial position y(0) = −2 and v(0) = 0.

Answer 1