Math 2250-4 Fri Jan 181.5: applications of linear DEs, including EP 3.7: electric circuitsIf time, begin section 2.1 improved population models. (There probably won't be much time for this, but it will be something to look forward to after the Monday holiday.)If there is interest, there will be a few more Maple introductions next week, TBA. By the way, you'll be submitting some computer work next week as part of your homework. Is there interest in more Maple sessions?Finish Wednesday's notes: Let's make sure we understand the basic input-output model, which is where we ended class on Wednesday. Proceed from there, showing that the 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUklbXN1YkdGJDYlRistRiM2JUY6Ri9GMi8lL3N1YnNjcmlwdHNoaWZ0R0Y9Rj4=has 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 then doing the application in Wednesday's notes about a polluted Lake Erie.EP 3.7 This is a supplementary section. If you bought the new book packet at the bookstore you have a copy. For those of you that don't have a hard copy, I've posted a .pdf on our homework page.Often the same DE can arise in completely different-looking situations. For example, first order linear DE's also arise (as special cases of second order linear DE's) in simple LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkUkxDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== circuit 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==coulombs accumulates on the capacitor, at a rate 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in the diagram above) amperes (coulombs/sec), i.e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGL0YyRjlGOS1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORmZuLUYsNiVRIklGJ0YvRjJGUEY5 .Kirchoff's Law: The sum of the voltage drops around any closed circuit loop equals the applied voltage LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== (volts). The units of voltage are energy units - Kirchoff's Law says that a test particle traversing any closed loop returns with the same potential energy level it started with:For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==: 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For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI6RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZURj0=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if no inductor, or if no capacitor, then Kirchoff's Law yields LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbWlHRiQ2JVEjc3RGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy order linear DE's, as below:Exercise 1: Consider the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJMRidGL0YyRjk= circuit below, in which a switch is thrown at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. Assume the voltage LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is constant, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj5GPg== = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv . Find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== . 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Improved population modelsLet LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== be a population at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . Let's call them "people", although they could be other biological organisms, decaying radioactive elements, accumulating dollars, or even molecules of solute dissolved in a liquid at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (2.1.23). Consider:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, birth rate (e.g. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJ3Blb3BsZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJXllYXJGJ0YyRjVGMkY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZCLyUpYmV2ZWxsZWRHUSZmYWxzZUYnL0Y2USdub3JtYWxGJw==) ; 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 , fertility rate (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJ3Blb3BsZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJXllYXJGJ0YyRjVGMkY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZCLyUpYmV2ZWxsZWRHUSZmYWxzZUYnL0Y2USdub3JtYWxGJw== per LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEncGVyc29uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== )LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjBRJXRydWVGJy9GM1EnaXRhbGljRidGMkYyRjI=, death rate (e.g. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJ3Blb3BsZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJXllYXJGJ0YyRjVGMkY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZCLyUpYmV2ZWxsZWRHUSZmYWxzZUYnL0Y2USdub3JtYWxGJw==); 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 , mortality rate (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJ3Blb3BsZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJXllYXJGJ0YyRjVGMkY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZCLyUpYmV2ZWxsZWRHUSZmYWxzZUYnL0Y2USdub3JtYWxGJw== per LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEncGVyc29uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== )Then in a closed system (i.e. no migration in or out) we can write the governing DE two equivalent ways: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 .Model 1: constant fertility and mortality rates, 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 , constants.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 .This is our familiar exponential growth/decay model, depending on whether LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= .Model 2: population fertility and mortality rates only depend on population LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, but they are not constant: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 constants. This 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 .For viable populations, 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. For a sophisticated (e.g. human) population we might also expect LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NDY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIjxGJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWS1GOzYkRkRGNUY1, and resource limitations might imply LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj5GJ0Y1LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWS1GOzYkRkRGNUY1. With these assumptions, and writing 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 one obtains the logistic differential equation:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZULUkobWZlbmNlZEdGJDYkLUYjNigtRiw2JVEiYkYnRi9GMi1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlxvLUYsNiVRImFGJ0YvRjItRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS0ZPRk1GK0Y5RjlGK0Zhb0Zhb0Y5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZULUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZaLUYsNiVRImFGJ0YvRjItRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS0ZPRk0tSSVtc3VwR0YkNiVGKy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjlGL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIRllGZW4tRiw2JVEiYkYnRi9GMkZpbkYrRjk= , or 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.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 . (One can consider other cases as well.)Exercise 2: Discuss qualitative features of the slope field for the logistic differential equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTEYrLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEidEYnRi9GMkY5RjlGOQ==: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZwcmltZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZULUYsNiVRImtGJ0YvRjItRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS0ZPRk1GKy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRIk1GJ0YvRjItRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZjb0YrRjlGOUY5a) There are two constant ("equilibrium") solutions. What are they?b) Evaluate the sign and magnitude of the slope function 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 , in order to understand and be able to recreate the two diagrams below. One is a qualititative picture of the slope field, in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJQRidGL0YyRjk= plane. The diagram to the left of it, called the phase diagram , is just a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= number line with arrows indicating whether LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== is increasing or decreasing on the intervals between the constant solutions. 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When discussing the logistic equation, the value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is called the "carrying capacity" of the (ecological or other) system. Discuss why this is a good way to describe LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= . Hint: if 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 , and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== solves the logistic equation, what is the apparent value of 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 ?Exercise 3: Solve the logistic DE IVPLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ki1GLDYlUSJQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJnByaW1lO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR1EmMC4wZW1GJy1GOzYtUSI9RidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjI3Nzc3NzhlbUYnL0ZTRlktRiw2JVEia0YnRjRGNy1GOzYtUSJ+RidGPkZARkNGRUZHRklGS0ZNL0ZQRlRGUkYxLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEiTUYnRjRGNy1GOzYtUSgmbWludXM7RidGPkZARkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTRmhvRjFGPkY+Rj5GK0Y+LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUklbXN1YkdGJDYlRistRiM2JUY6Ri9GMi8lL3N1YnNjcmlwdHNoaWZ0R0Y9Rj4=via separation of variables. Verify that the solution formula is consistent with the slope field and phase diagram discussion from exercise 1. Hint: You should find 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 .Solution (we will work this out step by step in class):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By partial fractions,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 .Use this expansion and multiply both sides of the separated DE by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to 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 .Integrate: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exponentiate: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Since the left-side is 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 (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdWJHRiQ2JUYrLUYjNiUtSSNtbkdGJDYkUSIyRidGOUYvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y5 or 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) (At LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= we see 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 .)Now, solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== by multiplying both sides of of the second to last equation by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==: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Collect LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== terms on left, and add LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 to both sides: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 .Plug in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and simplify: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 .Finally, because 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 , we see that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUknbXVuZGVyR0YkNiUtSSNtb0dGJDYtUSRsaW1GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2Ji1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dGQi9GM1EnaXRhbGljRictRi82LVEtJnJpZ2h0YXJyb3c7RidGMi9GNlEmZmFsc2VGJy9GOUZZL0Y7RkIvRj1GWS9GP0ZZL0ZBRlkvRkRGWS9GRlEsMC4yNzc3Nzc4ZW1GJy9GSUZbby1GLzYtUSgmaW5maW47RidGMkZYRlovRjtGWUZmbkZnbkZobkZpbkZFL0ZJRkdGMi8lLGFjY2VudHVuZGVyR0ZZLUZONiVRIlBGJ0ZRRlMtSShtZmVuY2VkR0YkNiQtRiM2JEZNRjJGMi1GLzYtUSI9RidGMkZYRlpGYG9GZm5GZ25GaG5GaW5Gam5GXG8tSSZtZnJhY0dGJDYoLUklbXN1YkdGJDYlLUZONiVRI01QRidGUUZTLUYjNiUtSSNtbkdGJDYkUSIwRidGMkZRRlMvJS9zdWJzY3JpcHRzaGlmdEdGXXEtRiM2JS1GY3A2JUZkb0ZocEZecUZRRlMvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmlxLyUpYmV2ZWxsZWRHRllGXHAtRk42JVEiTUYnRlFGU0Yy as expected.Note: If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RIj5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRldGOkY+ the denominator stays positive for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RLyZHcmVhdGVyRXF1YWw7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIwRidGOUY5, so we know that the formula for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==is a differentiable function for all LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. (If the denominator became zero, the function would blow up at the corresponding vertical asymptote.) To check that the denominator stays positive check that (i) if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RIjxGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctRi82JVEiTUYnRjJGNUY+ then the denominator is a sum of two positive terms; if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctRi82JVEiTUYnRjJGNUY+ the separation algorithm actually fails because you divided by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv to get started but the formula actually recovers the constant equilibrium solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSgmZXF1aXY7RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRIk1GJ0YvRjJGPQ==; and if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RIj5GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctRi82JVEiTUYnRjJGNUY+ then 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 so the second term in the denominator can never be negative enough to cancel out the positive LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+ , for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= .)Application! The Belgian demographer P.F. Verhulst introduced the logistic model around 1840, as a tool for studying human population growth. Our text demonstrates its superiority to the simple exponential growth model, and also illustrates why mathematical modelers must always exercise care, by comparing the two models to actual U.S. population data. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlNGTy1GNjYtUSIjRidGOUY7Rj5GQEZCRkRGRkZIRlJGVC1GLDYlUS5+Y2xlYXJ+bWVtb3J5RidGL0YyLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRlMvJSZkZXB0aEdGam4vJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGTy1GLDYlUSdEaWdpdHNGJ0YvRjItRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JI21uR0YkNiRRIjVGJ0Y5RjVGT0ZPRlUtRiw2JVE/d29ya353aXRofjV+c2lnbmlmaWNhbnR+ZGlnaXRzRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZDLUkjbWlHRiQ2JVElcG9wc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GUy1JKG1mZW5jZWRHRiQ2Ji1GIzZlcC1GVjYmLUYjNiYtSSNtbkdGJDYkUSUxODAwRidGOS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRlIvRk5RLDAuMzMzMzMzM2VtRictRmluNiRRJDUuM0YnRjlGOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRlxvLUZWNiYtRiM2Ji1GaW42JFElMTgxMEYnRjlGXG8tRmluNiRRJDcuMkYnRjlGOUY5RmVvRmhvRlxvLUZWNiYtRiM2Ji1GaW42JFElMTgyMEYnRjlGXG8tRmluNiRRJDkuNkYnRjlGOUY5RmVvRmhvRlxvLUZWNiYtRiM2Ji1GaW42JFElMTgzMEYnRjlGXG8tRmluNiRRJTEyLjlGJ0Y5RjlGOUZlb0Zob0Zcby1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0ZTLyUmZGVwdGhHRl5yLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZqcTYmRlxyRl9yRmFyL0ZkclElYXV0b0YnRk9GT0ZPRk9GT0ZPRk8tRlY2Ji1GIzYmLUZpbjYkUSUxODQwRidGOUZcby1GaW42JFElMTcuMUYnRjlGOUY5RmVvRmhvRlxvLUZWNiYtRiM2Ji1GaW42JFElMTg1MEYnRjlGXG8tRmluNiRRJTIzLjJGJ0Y5RjlGOUZlb0Zob0Zcby1GVjYmLUYjNiYtRmluNiRRJTE4NjBGJ0Y5RlxvLUZpbjYkUSUzMS40RidGOUY5RjlGZW9GaG9GXG8tRlY2Ji1GIzYmLUZpbjYkUSUxODcwRidGOUZcby1GaW42JFElMzguNkYnRjlGOUY5RmVvRmhvRlxvRmlxRmZyRk9GT0ZPRk9GT0ZPRk8tRlY2Ji1GIzYmLUZpbjYkUSUxODgwRidGOUZcby1GaW42JFElNTAuMkYnRjlGOUY5RmVvRmhvRlxvLUZWNiYtRiM2Ji1GaW42JFElMTg5MEYnRjlGXG8tRmluNiRRJTYzLjBGJ0Y5RjlGOUZlb0Zob0Zcby1GVjYmLUYjNiYtRmluNiRRJTE5MDBGJ0Y5RlxvLUZpbjYkUSU3Ni4yRidGOUY5RjlGZW9GaG9GXG8tRlY2Ji1GIzYmLUZpbjYkUSUxOTEwRidGOUZcby1GaW42JFElOTIuMkYnRjlGOUY5RmVvRmhvRlxvRmlxRmZyRk9GT0ZPRk9GT0ZPRk8tRlY2Ji1GIzYmLUZpbjYkUSUxOTIwRidGOUZcby1GaW42JFEmMTA2LjBGJ0Y5RjlGOUZlb0Zob0Zcby1GVjYmLUYjNiYtRmluNiRRJTE5MzBGJ0Y5RlxvLUZpbjYkUSYxMjMuMkYnRjlGOUY5RmVvRmhvRlxvLUZWNiYtRiM2Ji1GaW42JFElMTk0MEYnRjlGXG8tRmluNiRRJjEzMi4yRidGOUY5RjlGZW9GaG9GXG8tRlY2Ji1GIzYmLUZpbjYkUSUxOTUwRidGOUZcby1GaW42JFEmMTUxLjNGJ0Y5RjlGOUZlb0Zob0Zcb0ZpcUZmckZPRk9GT0ZPRk9GT0ZPLUZWNiYtRiM2Ji1GaW42JFElMTk2MEYnRjlGXG8tRmluNiRRJjE3OS4zRidGOUY5RjlGZW9GaG9GXG8tRlY2Ji1GIzYmLUZpbjYkUSUxOTcwRidGOUZcby1GaW42JFEmMjAzLjNGJ0Y5RjlGOUZlb0Zob0Zcby1GVjYmLUYjNiYtRmluNiRRJTE5ODBGJ0Y5RlxvLUZpbjYkUSYyMjUuNkYnRjlGOUY5RmVvRmhvRlxvLUZWNiYtRiM2Ji1GaW42JFElMTk5MEYnRjlGXG8tRmluNiRRJjI0OC43RidGOUY5RjlGZW9GaG9GXG9GaXFGZnJGT0ZPRk9GT0ZPRk9GTy1GVjYmLUYjNiYtRmluNiRRJTIwMDBGJ0Y5RlxvLUZpbjYkUSYyODEuNEYnRjlGOUY5RmVvRmhvRlxvLUZWNiYtRiM2Ji1GaW42JFElMjAxMEYnRjlGXG8tRmluNiRRJTMwOC5GJ0Y5RjlGOUZlb0Zob0Y5RjlGZW9GaG8tRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GT0ZPLUY2Ni1RIiNGJ0Y5RjtGPkZARkJGREZGRkhGUkZULUYsNiVRP0l+YWRkZWR+MjAxMH4tfmJldHdlZW5+MzA2LTMxM0YnRi9GMkZpcUZPRk9GT0ZPRk9GT0ZPRk9GYV5sLUYsNiVRaW5+SX51c2VkfnNoaWZ0LWVudGVyfnRvfmVudGVyfm1vcmV+dGhhbn5vbmV+bGluZX5vZn5pbmZvcm1hdGlvbkYnRi9GMkZpcUZPRk9GT0ZPRk9GT0ZPRk9GYV5sLUYsNiVRP35iZWZvcmV+ZXhlY3V0aW5nfnRoZX5jb21tYW5kLkYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRkM2LVEifkYnRkBGRkZIRkpGTEZORlBGUi9GVVEmMC4wZW1GJy9GWEZnbkZZLUZDNi1RIiNGJ0ZARkZGSEZKRkxGTkZQRlJGZm5GaG4tRiw2JVE+fnBsb3R0aW5nfmxpYnJhcnl+b2Z+Y29tbWFuZHNGJ0YvRjItSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGZ24vJSZkZXB0aEdGZG8vJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGWS1GLDYlUSpwb2ludHBsb3RGJ0YvRjItRjY2JC1GIzYqLUYsNiVRJXBvcHNGJ0YvRjItRkM2LVEiLEYnRkBGRi9GSUYxRkpGTEZORlBGUkZmbi9GWFEsMC4zMzMzMzMzZW1GJ0ZZLUYsNiVRJnRpdGxlRidGL0YyLUZDNi1RIj1GJ0ZARkZGSEZKRkxGTkZQRlJGVEZXLUYsNiVRP2BVLlMufnBvcHVsYXRpb25+dGhyb3VnaH50aW1lYEYnRi9GMkY9RkBGQC1GQzYtUSI7RidGQEZGRmlwRkpGTEZORlBGUkZmbkZXRj1GQA==JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Unlike Verhulst, the book uses data from 1800, 1850 and 1900 to get constants in our two models. We let t=0 correspond to 1800. Exponential Model: For the exponential growth model 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 we use the 1800 and 1900 data to get values for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRj4vJS9zdWJzY3JpcHRzaGlmdEdGPUZARj4= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnL0YzUSdub3JtYWxGJw== :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JSFHLogistic Model: We get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRj4vJS9zdWJzY3JpcHRzaGlmdEdGPUZARj4= from 1800, and use the 1850 and 1900 data to find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnL0YzUSdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnL0YzUSdub3JtYWxGJw== :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JSFHNow compare the two models with the real data, and discuss. The exponential model takes no account of the fact that the U.S. has only finite resources. Any ideas on why the logistic model begins to fail (with our parameters) around 1950?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JSFHJSFH