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Homework answers / question archive / Suppose the rate at which a rumor spreads - that is, the number of people who have heard the rumor over a period of time - increases with the number of people who have heard it

Suppose the rate at which a rumor spreads - that is, the number of people who have heard the rumor over a period of time - increases with the number of people who have heard it

Math

Suppose the rate at which a rumor spreads - that is, the number of people who have heard the rumor over a period of time - increases with the number of people who have heard it. If y is the number of people who have heard the rumor, the following function, where t is the time in days, is true. dy/dt=ky a) if y is 1 when t=0, and y is 3 when t=2, find k.

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K = 0.5493

The complete solution is given below.

Given that, dy Page -1 = ky = ) dy - - ky = 0 ( 1 ) Equation ( 1 ) is Lineal differential Equations . Compairing (1) with standard linear Equation. dy F + P y = Q we have , P = - K , Q = 0 Then, Integrating Factor ( I.F ) = pSpat = p J- kat = p- kt Now , Solution q ( 1) is , y x I . F = [ S Q X I . F } at + C -) yxekt = ofc = ) yxeut - C = ) 4 =cekt - (1 ) when 4 = 1 and t =0, then from (11) we have, 1 = cpk x o J = C. I [e = ] ] =) Cz1

Page - 2 Putting value q c in Equation ( 11 ) we get & = 1. ekt 2 ) y= ekt - (Ill ) Now put y = 3 and t = 2 in ( 111 ) we get 3 = p2 k =) In3 = 2K ( taking en bothsides ) =) 1 . 09 86 = 2K ( In 3 = 1. 0986 ) =) 1 . 0986 2 =) 0 . 5493 = K Therefore K = 0.5493

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