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Differential Equation Problem Help

  1. Sep 6, 2009 #1
    1. The problem statement, all variables and given/known data
    A certain piece of dubious information about phenylethylamine in the drinking water began to spread one day in a city with a population of 100,000. Within a week, 10,000 people had heard this rumor. Assume that the rate of increase of the number who have heard the rumor is proportional to the number who have not heard it. How long will it be until half the population of the city has heard the rumor?

    2. Relevant equations
    (Possibly): Natural Growth equation: dx/dt = kx (where k is a constant)

    3. The attempt at a solution
    I really have no clue how to do this.

    I've tried setting it up where dx/dt = k(100000 - x) and integrating, but I'm always left with two unknown constants.
  2. jcsd
  3. Sep 6, 2009 #2
    lmao sorry for wasting everyone's time, but I just worked out the answer myself. I'll explain it for those who are interested and then the thread can be locked or whatever.

    let x stand for the amount of the population that HAS heard the rumor
    it follows that 100000-x is then the number that HASN'T heard the rumor
    let dx/dt be the derivative of x with respect to time (in days)

    dx/dt = k(100000-x)
    where k is a constant

    Solving the separable differential equation leads to:

    x = 100000 - C*exp^(kt)
    where C is a constant

    Plug in the initial conditions:
    x(7) = 10000 = 100000 - C*exp^(k*7)
    which goes to:
    90000 = C*exp^(7k)

    I was screwing up before because I failed to assume another initial condition:
    Assume x(0) = 0:
    x(0) = 0 = 100000 - C*exp^(k*0)
    which leads to:

    Plugging C into: 90000 = C*exp^(7k) and solving for k gives:

    Then when you know all the constants, just solve for t in the eqn:

    50000 = 100000 - C*exp^(kt)

    t is about 46 days
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