chapter2.page12

chapter2.page12 - 2 , we have ( ax-bx 2 ) v + v x = axv-bx...

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12 1. EXPLICITLY SOLVABLE FIRST ORDER DIFFERENTIAL EQUATIONS Example 3.3 . In modelling a population with its births(per unit time) proportional to current population level and the deaths(per unit time) is proportional to the square of the current population, we have dx dt = ax - bx 2 , which is called logistic differential equation. It is a Bernoulli equation with n = 2 (Notice it is also a separable equation if a,b are constants.) Solution Let v = x 1 - 2 = x - 1 , then xv = 1 apply product rule of differentiation, we have x 0 v + v 0 x = 0, plug in x 0 = ax - bx
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Unformatted text preview: 2 , we have ( ax-bx 2 ) v + v x = axv-bx 2 v + v x = a-bx + v x = 0 since xv = 1 So a-bx + v x = 0 (7) Divide (7) by x and notice 1 x is v , we have v + av-b = 0 , or v + av = b. Apply the solution formula for linear equation we have v = e-R adt Z be R adt dt + C ! So x = 1 v = e R adt R be R adt dt + C ! a Example 3.4 . In the logistic model, dx dt = ax-bx 2 , suppose a > ,b > are constants, we will discuss behavior of the solutions....
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This note was uploaded on 10/04/2011 for the course MAP 4231 taught by Professor Dr.han during the Spring '11 term at UNF.

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