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Unformatted text preview: 2.2: Equilibrium Solutions and Stability Definitions Suppose that dx dt = f H x H t LL . Then L The solutions for f H x L = 0 are called critical numbers , c . L We say that x H t L ” c is an equilibrium solution . L Note that this is true because dc dt = = f H c L . L We say that the equilibrium solution is s t a b l e if for every Ε > 0, there exists Δ > 0, such that x- c < Δ x H t L- c < Ε , " t > 0. L Otherwise we say it is u n s t a b l e . Example Consider the differential equation dy dx = - 3 H y- 5 L = 3 H 5- y L . This is a Newton's Law of Cooling differential equation...where a body of initial temperature y H L = y is placed in an environment with temperature 5. The value k = - 3 is the constant of proportionality and the temperature of the body is proportional to the difference between its current temperature and that of the environment. L Find the critical numbers. L Note that the general solution is y H x L = 5 + A ª- 3 x . Determine the stability of c = 5. Note that here, y = y H L = 5 + A . L Illustrate the stability of c = 5 using a phase line diagram . theDE = 3 H 5- y L ; ivp = 88 0, 8 < , 8 0, 1 << ; Show @ VectorPlot @8 1, theDE < , 8 x,- 0.5, 1 < , 8 y,- 1, 10 < , FrameLabel fi 8 x, y < , Axes fi True, VectorScale fi 8 0.04, 0.2, None < , VectorStyle fi Gray, StreamScale fi Full, StreamStyle fi 8...
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This note was uploaded on 01/13/2012 for the course MATH 306 taught by Professor Keithemmert during the Spring '11 term at Tarleton.
- Spring '11