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Math Help - Plotting Difference Equation

  1. #1
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    Plotting Difference Equation

    Hey guys, I am currently trying to plot the graph of the equation
    N(t+1) = Lambda * N(t)

    Where Lambda is a constant.
    It is the basic unlimitted growth equation, and ideally i would like to plot it for different values of N(0) and different Lambda

    Only problem is, I have no idea how I plot this, do I need to solve it? if so how do i solve such an equation, i seem to remember somewhere that the solution would be N(t) = N(0) exp lambda*t

    is that correct?

    I am using the computer program Autograph, which lets me enter any equation (apart from ones like this :P) Hence I guess i need to solve it and plot the solution? Im kinda confusing myself.

    Many thanks for any help!

    Tom
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  2. #2
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    Quote Originally Posted by MWG_Thomas View Post
    Hey guys, I am currently trying to plot the graph of the equation
    N(t+1) = Lambda * N(t)

    Where Lambda is a constant.
    It is the basic unlimitted growth equation, and ideally i would like to plot it for different values of N(0) and different Lambda

    Only problem is, I have no idea how I plot this, do I need to solve it? if so how do i solve such an equation, i seem to remember somewhere that the solution would be N(t) = N(0) exp lambda*t

    is that correct?

    I am using the computer program Autograph, which lets me enter any equation (apart from ones like this :P) Hence I guess i need to solve it and plot the solution? Im kinda confusing myself.

    Many thanks for any help!

    Tom
    The solution to the functional equation N(t+1) = \lambda N(t) is N(t) = a \lambda^{t+1}.

    Edit: Should be N(t) = a \lambda^{t} - thanks for the catch, topsquark.
    Last edited by mr fantastic; March 26th 2008 at 04:38 PM. Reason: Fixed the latex. And noted an error.
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  3. #3
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    Thanks, how do i know what a is? and then once i have this a can i just plot it as you would any y=mx+c equation?

    Thanks again,

    Tom
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  4. #4
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    Quote Originally Posted by MWG_Thomas View Post
    Thanks, how do i know what a is? and then once i have this a can i just plot it as you would any y=mx+c equation?

    Thanks again,

    Tom
    a = N(0).
    Yes.
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  5. #5
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    N(t+1) = \lambda N(t) is N(t) = N(0) \lambda^{t+1}.

    So if that is my solution, why is it t+1 and not simply t?

    As say my N(0) is 1000 and lambda is 5 this equation would give when t=0
    is N(t) = 1000* 5^{1}. which = 5000, so then N(0) would be 5000? but i know that N(0) is 1000, so that can't be right?

    Isnt the solution
    N(t) = N(0) \lambda^{t}. This would then give me the correct answer for N(0)

    Thanx!!

    Tom
    Last edited by MWG_Thomas; March 26th 2008 at 07:34 AM.
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  6. #6
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    Quote Originally Posted by MWG_Thomas View Post
    N(t+1) = Lambda * N(t)
    N(t + 1) - \lambda N(t) = 0

    The characteristic equation for this is
    m - \lambda = 0

    m = \lambda

    Thus the solutions are of the form
    N(t) = a \lambda ^t

    where a = N(0).

    -Dan
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