# Show solutions are global

• Apr 4th 2009, 01:34 AM
rmangan
Show solutions are global
I need to show that al the solutions of x' = x + tcosx are global.

I think I understand what I have to do: I need to show that the solutions are defined on R.

I know that I need to construct barriers/fences to show this but I'm not sure how to do this for this de.

Thanks for any help.
• Apr 4th 2009, 11:06 AM
chisigma
Given a first order ODE...

$\displaystyle x^{'}= f(x,t)$ (1)

... conditioned by $\displaystyle x(t_{0})= x_{0}$, if in $\displaystyle [x_{0},t_{0}]$ both $\displaystyle f(*,*)$ and its partial derivative $\displaystyle f_{x}^{'}(*,*)$ exist and are continous, then there is one and only one solution of (1) so that $\displaystyle x(t_{0})=x_{0}$. In the case you have proposed is...

$\displaystyle f(x,t)= x + t\cdot \cos x$

... that is contionous in al the [x,t] plane, as well as...

$\displaystyle f_{x}^{'}(x,t)= 1 - t\cdot \sin x$

So all the solutions of...

$\displaystyle x^{'}= x + t\cdot \cos x$ (2)

... are global...

Kind regards

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