# Thread: Why bounded ode solution tends to a constant

1. ## Why bounded ode solution tends to a constant

Dear all, consider now a solution
$x(t):\ [0,\infty)\to R$
to an ode
$x'=f(x)$
If $x(t)$ is bounded. Show $\lim_{t\to\infty}x(t)=c$
for some constant $c$

2. ## Re: Why bounded ode solution tends to a constant

Originally Posted by xinglongdada
Dear all, consider now a solution
$x(t):\ [0,\infty)\to R$
to an ode
$x'=f(x)$
If $x(t)$ is bounded. Show $\lim_{t\to\infty}x(t)=c$
for some constant $c$
A counterxample: $x(t)= \sin t$ is bounded and obeys to ODE $x^{'}= \sqrt{1-x^{2}}$ , but the $\lim_{t \rightarrow \infty} x(t)$ doesn't exsist...

Kind regards

$\chi$ $\sigma$

3. ## Re: Why bounded ode solution tends to a constant

Originally Posted by xinglongdada
Dear all, consider now a solution $x(t):\ [0,\infty)\to R$ to an ode $x'=f(x)$ . If $x(t)$ is bounded. Show $\lim_{t\to\infty}x(t)=c$ for some constant $c$
Chisigma provided you an excellent answer. It would be interesting to transcribe the exact formulation of the problem.

4. ## Re: Why bounded ode solution tends to a constant

Indeed, this is a very nice counterexample...Good luck