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?

- Sep 8th 2009, 10:41 PMqtpipiBasic 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? - Sep 9th 2009, 04:09 AMchisigma
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

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