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# 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 field of the di ff 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. Figure 3 contains a plot of the slope field. We conclude that lim t →∞ y ( t ) = 0 whenever 0 < P < T , K whenever T < P . Figure 3. Slope Field for P = r P (1 P/K ) (1 P/T ) 0 0.8 1.6 2.4 3.2 4 4.8
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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 di±erential 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 di±erential 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 di±erentials such as dx and dy . Since this is the Diferential Calculus we can manipulate di±erentials in this way. Such an equation will also be called a frst order di±erential equation....
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