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MIT18_03S10_c08

# MIT18_03S10_c08 - 18.03 Class 8 Autonomous equations[1...

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18.03 Class 8 , Feb 19, 2010 Autonomous equations [1] Logistic equation [2] Phase line [3] Extrema, points of inflection Announcements: Final Tuesday, May 18, 9:00 - 12:00, Johnson Track Hour exam next Wednesday: Rooms to be announced Monday. Please go to the hour you are registered for. 50 minutes. [1] Back to qualitative study of differential equations. I'll use (t,y) today. The general first order equation is y'(t) = F(t,y) Autonomous ODE: y'(t) = g(y) . Autonomous means conditions are constant in time, though they may depend on the current value of y . Eg [Natural growth/decay] Constant growth rate: so y' = k0 y . Eg Bank account with interest rate NOT depending on time but possibly depending upon current balance, and constant savings rate: y' = I(y) y + q . Extended example: Population model with variable growth rate k(y) depending on the current population but NOT ON TIME; so y' = k(y) y. Suppose that when y is small the growth rate is approximately k0 , but that there is a maximal sustainable population p , and as y gets near to p the growth rate decreases to zero. When y > p , the growth rate becomes negative; the population declines back to the maximal sustainable population. In the simplest version of this, when you graph of k(y) against y you get a straight line with vertical intercept k0 and horizontal intercept p : k(y) = k0 (1 - (y/p)). so k(0) = k0 , and k(p) = 0 . When y > p , k(y) < 0 . The Logistic Equation is y' = k0 (1 - (y/p)) y = g(y) . This is more realistic than Natural Growth when you want to account for limits to growth. It is nonlinear but it IS autonomoous.

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MIT18_03S10_c08 - 18.03 Class 8 Autonomous equations[1...

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