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# Advanced Math/Systems of Autonomous Equations - Kermack-McKendrick Model

Question
Hi Clyde,

I have an extremely hard calculus question as shown in the picture attached and I don't quite know how to approach it. The question I am stuck on is question #4 of 8.3.1.

Thank you very much for all your help,
Sam

autonomous NL eqs
Sam,

The attached pdf file shows the solution. This problem really isn't extremely hard, as you'll see. There are just a couple of "tricks" you need to use to wrangle the equtions into the desired form.

Perhaps the biggest "trick" is using the fact that the non-linear (NL) equation is SEPARABLE. This makes the solution much easier. Not all NL eqs have this property so be careful. If an eqn is not separable, then you may need to see if it is EXACT, in which case there is a straightforward technique for solving it. If it is not exact to start with, you may be able to derive an exact eqn using an INTEGRATING FACTOR, which is more involved but still ends up being mainly a crank-turning exercise.

It is also fortunate that the eqn is AUTONOMOUS, which means the independent variable t doesn't appear explicitly.

Randy (not Clyde)
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#### randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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