lecture_9 (dragged) 2 - 0.8 1.6 2.4 3.2 4 4.8 5.6 6.4 7.2 8...

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MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 2 3 depends on the initial value P (0) = P 0 , we consider the slope feld oF the di±erential equation. The three critical points P L =0 ,T,K break the graph into Four regions, but we just consider three oF them: When 0 <P<T we have f ( P ) < 0, so that P = P ( t ) is a decreasing Function. This is because P<T is below the threshold so we expect the population to become extinct. When T<P<K we have f ( P ) > 0, so that P = P ( t ) is an increasing Function. This is because P<K so the population is below the carrying capacity oF the population. When P>K we have f ( P ) < 0, so that P = P ( t ) is a decreasing Function. This is because P is above the carrying capacity oF the population. ²igure 3 contains a plot oF the slope feld. We conclude that lim t →∞ y ( t )= ° 0w h e n e v e r 0 <P<T , K whenever T<P . Figure 3. Slope ²ield For P ° = rP
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Unformatted text preview: 0.8 1.6 2.4 3.2 4 4.8 5.6 6.4 7.2 8 8.8 9.6 10.4-15-10-5 5 10 15 20 25 30 35 Exact Equations First Order Equations Revisited. Recall that every frst order dierential equation is in the Form dy dx = G ( x,y ) For some Function G = G ( x,y ). Say that there are two Functions M = M ( x,y ) and N = N ( x,y ) such that G ( x,y ) = M ( x,y ) N ( x,y ) . Then we can write the frst order dierential equation in the Form dy dx = M ( x,y ) N ( x,y ) = M ( x,y ) dx + N ( x,y ) dy = 0 . Recall that a diferential equation is simply an equation involving dierentials such as dx and dy . Since this is the Diferential Calculus we can manipulate dierentials in this way. Such an equation will also be called a frst order dierential equation....
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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