Notes-1st order ODE pt2 - Autonomous Equations / Stability...

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Unformatted text preview: Autonomous Equations / Stability of Equilibrium Solutions First order autonomous equations, Equilibrium solutions, Stability, Long- term behavior of solutions, direction fields, Population dynamics and logistic equations Autonomous Equation : A differential equation where the independent variable does not explicitly appear in its expression. It has the general form of y = f ( y ). Examples : y = e 2 y- y 3 y = y 3- 4 y y = y 4- 81 + sin y Every autonomous ODE is a separable equation. Because ) ( y f dt dy = dt y f dy = ) ( = dt y f dy ) ( . Hence, we already know how to solve them. What we are interested now is to predict the behavior of an autonomous equations solutions without solving it, by using its direction field. Equilibrium solutions Equilibrium solutions (or critical points) occur whenever y = f ( y ) = 0. That is, they are the roots of f ( y ). Equilibrium solutions are constant functions that satisfy the equation, i.e. they are the constant solutions of the differential equation. Example : Logistic Equation of Population 2 1 y K r ry y K y r y- = - = Both r and K are positive constants. The solution y is the population size of some ecosystem, r is the intrinsic growth rate , and K is the environmental carrying capacity . The intrinsic growth rate is the natural rate of growth of the population provided that the availability of necessary resource (food, water, oxygen, etc) is limitless. The environmental carrying capacity (or simply, carrying capacity) is the maximum sustainable population size given the actual availability of resource. Without solving this equation, we will examine the behavior of its solution. Its direction field is shown in the next figure. Notice that the long-term behavior of a particular solution is determined solely from the initial condition y ( t ) = y . The behavior can be categorized by the initial value y : If y < 0, then y as t . If y = 0, then y = 0, a constant/equilibrium solution. If 0 < y < K , then y K as t . If y = K , then y = K , a constant/equilibrium solution. If y > K , then y K as t . Comment : In a previous section (applications: air-resistance) you learned an easy way to find the limiting velocity without having to solve the differential equation. Now we can see that the limiting velocity is just the equilibrium solution of the motion equation (which is an autonomous equation). Hence it could be found by setting v = 0 in the given differential equation and solve for v . Stability of an equilibrium solution The stability of an equilibrium solution is classified according to the behavior of the integral curves near it they represent the graphs of particular solutions satisfying initial conditions whose initial values, y , differ only slightly from the equilibrium value....
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This note was uploaded on 07/16/2011 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.

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Notes-1st order ODE pt2 - Autonomous Equations / Stability...

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