Globally defined solutions.
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From: Joubert on 25 Jun 2010 00:41 Take a Cauchy problem like: y'=xsiny y(0)=y* Show that the problem has a unique solution globally defined on R. I'm having trouble with this type of questions: how does one prove global uniqueness? Only via the usual theorems? How does one show a solution is "globally" defined on R? Another example is y'=1/y - 1/x y(1)=1 with a solution that should be globally defined on (0,+infinity). Any insight is helpful.
From: Robert Israel on 25 Jun 2010 01:06 Joubert <trappedinthecloset9985(a)yahoo.com> writes: > Take a Cauchy problem like: > > y'=xsiny > y(0)=y* > > Show that the problem has a unique solution globally defined on R. > I'm having trouble with this type of questions: how does one prove > global uniqueness? Only via the usual theorems? How does one show a > solution is "globally" defined on R? The only way for the solution to not be globally defined and unique (if the right side is everwhere continuously differentiable) is for it to go to infinity. Use inequalities to estimate how fast |y(x)| can grow. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Joubert on 25 Jun 2010 09:16 On 25/06/2010 7.06, Robert Israel wrote: > The only way for the solution to not be globally defined and unique > (if the right side is everwhere continuously differentiable) is for > it to go to infinity. Use inequalities to estimate how fast |y(x)| can > grow. You mean for the right hand side to go to infinity? So for instance considering y'=f(x,y) if lim x->+infty f(x,y) = infty then the solution is not globally defined? Why?
From: Robert Israel on 25 Jun 2010 19:27 Joubert <trappedinthecloset9985(a)yahoo.com> writes: > On 25/06/2010 7.06, Robert Israel wrote: > > > The only way for the solution to not be globally defined and unique > > (if the right side is everwhere continuously differentiable) is for > > it to go to infinity. Use inequalities to estimate how fast |y(x)| can > > grow. > > You mean for the right hand side to go to infinity? So for instance > considering > > y'=f(x,y) > > if lim x->+infty f(x,y) = infty then the solution is not globally > defined? Why? No, I mean for |y(x)| to go to infinity as x approaches a finite value. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Joubert on 25 Jun 2010 20:27
On 26/06/2010 1.27, Robert Israel wrote: > No, I mean for |y(x)| to go to infinity as x approaches a finite value. Then take for instance y'=(y^2 - 1)(y^2 + x^2) y(0)=y* I know for sure that if y* > 1 there is not a globally defined solution (as opposed to when |y*| < 1) but why? It doesn't look like the right hand member ever goes to infinity (except when y does). |