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Differential Equation Problem

  1. Oct 20, 2015 #1
    1. The problem statement, all variables and given/known data
    This is a interesting (morbid) problem from Simmons- Calculus with Analytic Geometry.
    In a certain barbourous land, two neighbouring tribes have hated one another from time immemorial. Being barbourous peoples, their powers of belief are strong, and a solemn curse pronounced by the medicine man of the first tribe deranges and drives them to murder and suicide. If the rate of change of the population P of the second tribe is ##-\sqrt{P}## per week, and if the population is 676 when the curse is uttered, when will they all be dead?

    Intial Conditions
    ##P(0) = 676##

    2. Relevant equations

    None

    3. The attempt at a solution

    ##\frac{dP}{dt} = -\sqrt{t} = -t^{1/2} ##.

    Separating the differential equation,

    ##dP = -t^{1/2}dt ##,

    Then, by intergrating,

    ##\int dP = - \int t^{1/2}dt ##

    ##P = -\frac{2}{3} t^{3/2}+ C ##......(1)

    Solving for C at t= (0) or P(0) weeks, when the medicine man uttered his curse,

    ##676 = -\frac{2}{3} (0)^{3/2} + C ##,
    ##C = 676##.

    Subbing this in (1)

    ##P = -\frac{2}{3} t^{3/2}+ 676 ## ......(2)

    Rearanging (2) for t, when P = 0 because the second tribe are all dead,

    ##0 = -\frac{2}{3} t^{3/2}+ 676##,
    ##-676 = -\frac{2}{3} t^{3/2}##,
    ##t = (\frac{2028}{2})^{2/3}= 100.93 = 101## weeks .

    Is this correct. Or have I made a massive error? Seems like they need a more powerful medicine man...
     
    Last edited: Oct 20, 2015
  2. jcsd
  3. Oct 20, 2015 #2

    SteamKing

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    Yes, you have made a massive error.

    According to the OP, "the rate of change of the population P of the second tribe is ##-\sqrt{P}## per week", yet you have set your ODE = ##-\sqrt{t}##. Why is that?
     
  4. Oct 20, 2015 #3

    hunt_mat

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    No, the equation you need is [tex]\frac{dP}{dt}=-\sqrt{P}[/tex]
     
  5. Oct 20, 2015 #4
    Dang. I had this on my first attempt but it just looked wrong and unfamiliar, so played around with the equation, this was the first attempt,

    ##\frac{dP}{dt} = -\sqrt{P} = -P^{1/2} ##.
    and rearanged it to,

    ##t = \int \frac{1}{-p^{1/2}}dP = ##.

    Is that ok?
     
  6. Oct 20, 2015 #5

    SteamKing

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    Yep, that's what you should start with.
     
  7. Oct 20, 2015 #6
    Ok this is not working out,
    ##t = -2\sqrt{P} + C##
    C = 52
    ##t = -2\sqrt{P} + 52##
    ##t = 0.##
     
  8. Oct 20, 2015 #7
    This is the correct solution. Substitute P=0 to find t.
     
  9. Oct 20, 2015 #8
    52 weeks... not a bad medicine man after all. I on the other hand, need to work more on ode magick.
     
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