Solution: 5055. Equilibrium solutions A differential equation of the form y 1t2 = F 1y2

Chapter 7, Problem 54

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5055. Equilibrium solutions A differential equation of the form y 1t2 = F 1y2 is said to be autonomous (the function F depends only on y). The constant function y = y0 is an equilibrium solution of the equation provided F 1y02 = 0 (because then y 1t2 = 0, and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of t. Consider the following equations. a. Find all equilibrium solutions. b. Sketch the direction field on either side of the equilibrium solutions for t 0. c. Sketch the solution curve that corresponds to the initial condition y102 = 1. y _1t2 = sin y