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Thread: Basic Diff EQ question: finding a function with given properties

  1. #1
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    Basic Diff EQ question: finding a function with given properties

    Find a function f(y) with the property that: for each solution y(x) of dy/dx = f(y), the limiting value lim as x-> +inf of y(x) equals 3 if y(0) > 0.

    Any help?
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  2. #2
    MHF Contributor chisigma's Avatar
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    A possible solution of the problem is a function $\displaystyle y(x)$ with the following derivative $\displaystyle y^{'} (x)$...

    $\displaystyle y^{'} < 0 $ with $\displaystyle y>3$ or $\displaystyle y<0$

    $\displaystyle y^{'} =0$ with $\displaystyle y=3$ or $\displaystyle y=0$

    $\displaystyle y^{'} >0$ with $\displaystyle 0<y<3 $

    A DE the solution of which has these properties is, among others, the following ...

    $\displaystyle y^{'} = 3 y - y^{2}, y(0)=y_{0}>0$

    ... as you can easily verify...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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