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Math Help - Doomsday Equation Example (Can you check my work?)

  1. #1
    Junior Member Coop's Avatar
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    Question Doomsday Equation Example (Can you check my work?)

    Hi,

    I just wanted to make sure that I completed this problem correctly, because I had a little bit of a rough time with it.

    The question reads,

    "Let c be a positive number. A differential equation of the form

    (dy/dt) = ky^(1+c)

    where
    k is a positive constant, is called a doomsday equation because the exponent in the expression ky^(1+c) is larger than the exponent 1 for natural growth."

    c) An especially prolific breed of rabbits has the growth term ky^(1.01). If two such rabbits breed initially and the warren has 16 rabbits after three months, when is the doomsday?

    Here's what I did...

    (dy/dt) = ky^(1.01); c = .01

    Separating the DE I get...

    (integral) dy/(y^1.01) = k integral (dt)

    Integrating...

    (y^-.01)/-.01 = kt + A <- "A" is my constant

    Now I plug in the initial condition y(0) = 2

    (2^-.01)/-.01 = k(0) + A

    A = (2^-.01)/-.01 = -99.3

    Now I plug in y(3) = 16 and A = -99.3

    (16^-.01)/-.01 = k(3) + -99.3

    Solving for k I get...

    [ (16^-.01)/-.01 + 99.3] / 3 = k = .681

    Now I plug k and A back into the original equation...

    (y^-.01)/-.01 = .681(t) + -99.3

    The question is asking me when the lim(t -> T) = infinity, right? So that means there is an asymptote at t = T. This means I need to find a t that y(t) does not occupy.

    Solving for y...

    y^-.01 = (.681(t) + -99.3)*-.01

    y^-.01 = -.00681t + .993

    y = 1/(-.00681t + .993)^.01

    So if I solve -.00681t + .993 = 0 for t, then that should be my answer, right? Because that would mean there is a 0 in the denominator.

    t = -.993 / -.00681 = (approx) 146 months

    Is what I did correct?

    Thanks!

    P.S. Can anyone tell me the units for k and A?
    Last edited by Coop; February 19th 2013 at 08:41 AM.
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  2. #2
    MHF Contributor

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    Re: Doomsday Equation Example (Can you check my work?)

    Quote Originally Posted by Coop View Post
    Hi,

    I just wanted to make sure that I completed this problem correctly, because I had a little bit of a rough time with it.

    The question reads,

    "Let c be a positive number. A differential equation of the form

    (dy/dt) = ky^(1+c)

    where
    k is a positive constant, is called a doomsday equation because the exponent in the expression ky^(1+c) is larger than the exponent 1 for natural growth."

    c) An especially prolific breed of rabbits has the growth term ky^(1.01). If two such rabbits breed initially and the warren has 16 rabbits after three months, when is the doomsday?

    Here's what I did...

    (dy/dt) = ky^(1.01); c = .01

    Separating the DE I get...

    (integral) dy/(y^1.01) = k integral (dt)

    Integrating...

    (y^-.01)/-.01 = kt + A <- "A" is my constant

    Now I plug in the initial condition y(0) = 2

    (2^-.01)/-.01 = k(0) + A

    A = (2^-.01)/-.01 = -99.3
    Yes, that's good.

    Now I plug in y(3) = 16 and A = -99.3

    (16^-.01)/-.01 = k(3) + -99.3

    Solving for k I get...

    [ (16^-.01)/-.01 + 99.3] / 3 = k = .681

    Now I plug k and A back into the original equation...

    (y^-.01)/-.01 = .681(t) + -99.3
    Okay

    The question is asking me when the lim(t -> T) = infinity, right? So that means there is an asymptote at t = T. This means I need to find a t that y(t) does not occupy.

    Solving for y...

    y^-.01 = (.681(t) + -99.3)*-.01

    y^-.01 = -.00681t + .993

    y = 1/(-.00681t + .993)^.01
    No. y= 1/(-.00681t+ .993)^(1/.01)= 1/(-.00681t+ .993)^(100).

    So if I solve -.00681t + .993 = 0 for t, then that should be my answer, right? Because that would mean there is a 0 in the denominator.

    t = -.993 / -.00681 = (approx) 146 months

    Is what I did correct?
    Except for the ".01" power where it should be 100, yes. And that does not affect the final answer, the number of months.

    Thanks!

    P.S. Can anyone tell me the units for k and A?[/QUOTE]
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  3. #3
    Junior Member Coop's Avatar
    Joined
    Feb 2013
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    New York, USA
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    Re: Doomsday Equation Example (Can you check my work?)

    "Except for the ".01" power where it should be 100, yes. And that does not affect the final answer, the number of months."
    ---

    Ah of course, thanks!
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