Solving DE for the spread of a rumour.

HallsofIvy

MHF Helper
Apr 2005
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The integral of \(\displaystyle \frac{dx}{x(1- x)}\) is NOT "ln|x(1- x)|". First write it as "partial fractions": \(\displaystyle \frac{1}{x(1-x)}= \frac{A}{x}+ \frac{B}{1- x}\), then integrate.
 

Prove It

MHF Helper
Aug 2008
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here is my problem and what I have done so far.
\(\displaystyle \frac{dx}{dt} = x(1 - x)\)

\(\displaystyle \frac{dt}{dx} = \frac{1}{x(1 - x)}\).


Now using the method of partial fractions:

\(\displaystyle \frac{A}{x} + \frac{B}{1 - x} = \frac{1}{x(1 - x)}\)

\(\displaystyle \frac{A(1 - x) + Bx}{x(1 - x)} = \frac{1}{x(1 - x)}\)

\(\displaystyle A(1 - x) + Bx = 1\)

\(\displaystyle A - Ax + Bx = 1\)

\(\displaystyle A + (B - A)x = 1 + 0x\).


Therefore \(\displaystyle A = 1\) and \(\displaystyle B - A = 0\), so \(\displaystyle B = 1\).

Thus \(\displaystyle \frac{1}{x(1 - x)} = \frac{1}{x} + \frac{1}{1 - x}\).



Back to the DE:

\(\displaystyle \frac{dt}{dx} = \frac{1}{x(1 - x)}\)

\(\displaystyle \frac{dt}{dx} = \frac{1}{x}+ \frac{1}{1 - x}\)

\(\displaystyle t = \int{\left(\frac{1}{x} + \frac{1}{1 - x}\right)\,dx}\)

\(\displaystyle t = \ln{|x|} - \ln{|1 - x|} + C\)

\(\displaystyle t = \ln{\left|\frac{x}{1 - x}\right|} + C\)

\(\displaystyle t = \ln{\left|\frac{x - 1 + 1}{1 - x}\right|} + C\)

\(\displaystyle t = \ln{\left|\frac{-(1 - x)}{1 - x} + \frac{1}{1 - x}\right|} + C\)

\(\displaystyle t = \ln{\left|-1 + \frac{1}{1 - x}\right|} + C\)

\(\displaystyle t - C = \ln{\left|-1 + \frac{1}{1 - x}\right|}\)

\(\displaystyle e^{t - C} = \left|-1 + \frac{1}{1 - x}\right|\)

\(\displaystyle e^{-C}e^t = \left|-1 + \frac{1}{1 - x}\right|\)

\(\displaystyle \pm e^{-C}e^t = -1 + \frac{1}{1 - x}\)

\(\displaystyle A\,e^t = -1 + \frac{1}{1 - x}\), where \(\displaystyle A = \pm e^{-C}\)

\(\displaystyle A\,e^t + 1 = \frac{1}{1 - x}\)

\(\displaystyle \frac{1}{A\,e^t + 1} = 1 - x\)

\(\displaystyle x = 1 - \frac{1}{A\,e^t + 1}\)

\(\displaystyle x = \frac{A\,e^t + 1 - 1}{A\,e^t + 1}\)

\(\displaystyle x = \frac{A\,e^t}{A\,e^t + 1}\).
 
May 2010
4
0
thank you.

and C is? .
 
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Prove It

MHF Helper
Aug 2008
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the first A has nothing to do with a second A. It is confusing. should i have used a different letter for that constant at the end of the calculation.
They are the same \(\displaystyle A\). Otherwise I would have used different letters.
 
May 2010
4
0
I am sorry that I am so stupit. It is not ez trust me

I am sorry, but I do not understand how did the first A, just a constant initially, all the sudden became +-e^(-c).
 
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May 2010
4
0
If we use this DE solution, x=1/(1-e^(-t)) and try to find the proportion of the population that has heard the rumor at the time t=0. The solution gives us the result - 0.5; that is a half of population before the rumor has started spreading? It doesn't make sense. Am I interpreting it wrong?
 
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mr fantastic

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Dec 2007
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I am sorry that I am so stupit. It is not ez trust me

I am sorry, but I do not understand how did the first A, just a constant initially, all the sudden became +-e^(-c).
C is arbitrary therefore -C is arbitrary therefore e^(-C) is arbitrary therefore it can be represneted by a new arbitrary symbol eg. A.
 

mr fantastic

MHF Hall of Fame
Dec 2007
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If we use this DE solution, x=1/(1-e^(-t)) and try to find the proportion of the population that has heard the rumor at the time t=0. The solution gives us the result - 0.5; that is a half of population before the rumor has started spreading? It doesn't make sense. Am I interpreting it wrong?
1. The given solution is x=1/(1+e^(-t)), not what you have said.

2. t = 0 => x = 1/2. All that means is that at t = 0 half the population have heard the rumour. Big deal.

3. The question asked you to show that the solution was x=1/(1+e^(-t)). So you don't actually have to solve the DE. Just substitute x=1/(1+e^(-t)) into it and show that the resulting left hand and right hand sides are equal to each other. The fact that you have been given no boundary condition suggests that this is the approach you were meant to take ... (And given what I have said in my second point, a possible boundary condition would have been the initial condition x(0) = 1/2).