Show solutions are global

**rmangan** · April 4th 2009, 01:34 AM

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.

**chisigma** · April 4th 2009, 11:06 AM

Given a first order ODE...

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

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

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

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

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

So all the solutions of...

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

... are global...

Kind regards

$\chi$ $\sigma$

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