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Math Help - Basis of Attraction (differential eq'ns)

  1. #1
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    Basis of Attraction (differential eq'ns)

    Consider the IVP given by
    y' = (1-y)^2 y(0) = yo

    For each equilibrium solution, determine the basin of attraction.

    I found the only equilibrium solution, y=1, but can't find the basis of attraction.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Separating the variables we have...

    \int \frac{dy}{(1-y)^{2}} = \int dx \rightarrow \frac {1}{1-y} = x + c \rightarrow y= \frac{x+c-1}{x+c}

    If we impose the condition  y(0)= y_{0} we obtain...

    c=\frac{1}{1-y_{0}}

    If y_{0}<1 then c>0 and ...

    \lim_{x \rightarrow \infty} y(x) = 1

    ... and the 'basis of attraction' is y=1 .

    If y_{0}>1 however the situation is quite different and y=1 becomes 'basis of repulsion'...

    Kind regards

    \chi \sigma
    Last edited by chisigma; July 28th 2009 at 11:14 PM. Reason: some imprecisions corrected
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  3. #3
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    Thanks chisigma.

    The question defines the basis of attraction as follows...

    "Basins of Attraction:
    Suppose that y=c is an equilibrium solution or constant solution of the first-order DE y'=f(y).
    The basin of attraction is the set of initial conditions y(0)=yo so that the solutions satisfying the IVP y'=f(y), y(0)=yo tend to c as t-->infinity."

    So in other words, the basis of attraction is the set of initial conditions such that the function will be attracted to a given equilibrium solution c.

    I approached this problem by graphing the DE y' = (1-y)^2 using this online app DiffEqu.

    I noted the equilibrium solution y=1, which my online homework says is correct.
    I then inputted the basis of attraction as (-inf,1), because all points contained in this basis of attraction tend towards y=1 as t approaches infinity.
    I'm pretty sure that this is the correct answer, but my online homework says it's incorrect.

    Perhaps I am entering it in the wrong format.

    From my homework...
    "Rules for inputting answer:
    (1) Start with the smallest equilibrium solution and input these solutions in increasing order.
    (2) If the basin of attraction is the point c, input only the number c without any spaces.
    (3) If the basin of attraction is an interval, input the interval without any spaces. For example, input can look like (-5,7) for an interval. Use -inf for negative infinity and inf for positive infinity.
    (4) If any answer fields are unused, type only an upper-case N in each of these."

    I inputted (-inf,1), but this is incorrect, still unsure where my error lies.
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  4. #4
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    Quote Originally Posted by FZ44 View Post
    Thanks chisigma.

    The question defines the basis of attraction as follows...

    "Basins of Attraction:
    Suppose that y=c is an equilibrium solution or constant solution of the first-order DE y'=f(y).
    The basin of attraction is the set of initial conditions y(0)=yo so that the solutions satisfying the IVP y'=f(y), y(0)=yo tend to c as t-->infinity."

    So in other words, the basis of attraction is the set of initial conditions such that the function will be attracted to a given equilibrium solution c.

    I approached this problem by graphing the DE y' = (1-y)^2 using this online app DiffEqu.

    I noted the equilibrium solution y=1, which my online homework says is correct.
    I then inputted the basis of attraction as (-inf,1), because all points contained in this basis of attraction tend towards y=1 as t approaches infinity.
    I'm pretty sure that this is the correct answer, but my online homework says it's incorrect.

    Perhaps I am entering it in the wrong format.

    From my homework...
    "Rules for inputting answer:
    (1) Start with the smallest equilibrium solution and input these solutions in increasing order.
    (2) If the basin of attraction is the point c, input only the number c without any spaces.
    (3) If the basin of attraction is an interval, input the interval without any spaces. For example, input can look like (-5,7) for an interval. Use -inf for negative infinity and inf for positive infinity.
    (4) If any answer fields are unused, type only an upper-case N in each of these."

    I inputted (-inf,1), but this is incorrect, still unsure where my error lies.
    I agree with you. Also, your online direction field plotter supports that.
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