c08 - 18.03 Class 8, Feb 24, 2006 Autonomous equations I'll...

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18.03 Class 8, Feb 24, 2006 Autonomous equations I'll use (t,y) today. y' = F(t,y) is the general first order equation Autonomous ODE: y' = g(y) . Eg [Natural growth/decay] Constant growth rate: so y' = k0 y . k0 > 0 means the populuation (if positive) is growing; k0 < 0 means it is falling, decaying. Autonomous means conditions are constant in time, though they may depend on the current value of y . Eg [Logistic equation] 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. Simplest version: A graph of k(y) against y , straight line with vertical intercept k0 and horizontal intercept p : k(y) = k0 (1 - (y/p)). so k(0) = k0 , and k(p) = 0 . The Logistic Equation is y' = k0 (1 - (y/p)) y = g(y) . This is more realistic than Nat Growth for large populations. It is nonlinear. Autonomous equations are always separable, but we aim for a qualitative grasp of solutions. Sketch direction field, isoclines first. Values of
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c08 - 18.03 Class 8, Feb 24, 2006 Autonomous equations I'll...

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