Find the function f(x) such that f\'(x) = f(x)(1 - f(x)) and f(0) = 1/8. Use f f

Question

Find the function f(x) such that f'(x) = f(x)(1 - f(x)) and f(0) = 1/8. Use f for f(x) in your equation.) An integral equation is an equation that contains an unknown function y(x) and an integral that involves y(x). Solve the given integral equation. y(x) = 2 + integral^x_4 [t - ty(t)] dt This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. The differential equation below models the temperature of a 88 degree C cup of coffee in a 20 degree C room, where it is

Explanation / Answer

8)

given f '(x)= f(x)(1-f(x)) , f(0)=1/8

df/dx =f(1-f)

seperate the variables

df/f(1-f) =dx

[(1/f)+(1/(1-f))]df =dx

inttegrate on both sides

[(1/f)+(1/(1-f))]df =dx

ln(f) -ln(1-f) =x +c

ln(f/(1-f)) =x +c

(f/(1-f)) =ex +c

(f/(1-f)) =Cex

((1-f)/f) =Ce-x

(1/f)-1=Ce-x

(1/f)=1+Ce-x

f=1/(1+Ce-x)

so f(x)=1/(1+Ce-x)

given f(0)=1/8

1/8=1/(1+Ce-0)

1/8=1/(1+C)

1+C=8

C=7

therefore function  f(x)=1/(1+7e-x) or  f(x)=ex/(ex+7)

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