c08mudd - 18.03 Muddy Card responses Thank you all for your...

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18.03 Muddy Card responses, February 24, 2006 Thank you all for your frank responses. I’ll try to answer some of the most common confusions. 1. A first order ODE is “autonomous” if it has the form y ˙ = g ( y ). Here ˙ y = dy/dt . y is a function of t , so it’s true that the right hand side depends on t ; but only through the dependence of y on t . y ˙ = ty is not autonomous; ˙ y = y is. The function g ( y ) completely describes the differential equation. In the natural growth/decay case, g ( y ) = k 0 y . In the logistic case, g ( y ) = k 0 (1 ( y/p )) y or, with harvest, g ( y ) = k 0 (1 ( y/p )) y a . If you explore the Phase lines Mathlet you will see other examples. 2. There were some questions about how I derived these formulas for g ( y ). Even in the natural growth case, when g ( y ) = k 0 y , someone made the good point that the growth rate k 0 reflects not only the birth rate but also the death rate. In fact it’s the birth rate minus the death rate; the two processes act against each other.
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