# Thread: Prove that y goes to negative infinity

1. ## 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 $x(t)>0$ be a continuous solution to the equation

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

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

and let $y=x+h_2$.

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

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

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

(3) $y>h_2$

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

Maybe these are useful, maybe not.

Any help would be much appreciated. Thanks!