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Thread: Prove that y goes to negative infinity

  1. #1
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    Prove that y goes to negative infinity

    Hi guys. I've got what *should* be a simple problem, but it's giving me trouble. Here it is:

    Let $\displaystyle x(t)>0$ be a continuous solution to the equation

    $\displaystyle x^{(4)}(t)+q(t)f(x(t))=-h_2^{(4)}(t)$,

    where $\displaystyle q>0$ and $\displaystyle xf(x)>0$ for $\displaystyle x\neq 0$, with $\displaystyle q,f,h_2$ continuous.

    and let $\displaystyle y=x+h_2$.

    Show that if $\displaystyle y^{(3)}\leq 0$ for arbitrarily large $\displaystyle t$ then $\displaystyle y\to-\infty$ as $\displaystyle t\to\infty$.
    It is easy to deduce the following results:

    (1) $\displaystyle y^{(4)}+qf(y-h_2)=0$

    (2) $\displaystyle y^{(4)}<0$

    (3) $\displaystyle y>h_2$

    (4) $\displaystyle y^{(3)}$ is strictly decreasing.

    Maybe these are useful, maybe not.

    Any help would be much appreciated. Thanks!
    Last edited by hatsoff; Jun 11th 2011 at 08:01 AM.
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