Very vicious differential problem
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From: Joubert on 10 Jun 2010 14:43 a > 0, b real. y' = ysin(y)/(1+x^a) lim (for x -> infty) y(x) = b Existence and uniqueness of C^1 solutions is the matter. This seems even harder than the previous.
From: Robert Israel on 10 Jun 2010 15:18 On Thu, 10 Jun 2010 20:43:29 +0200, Joubert wrote: > a > 0, b real. > > y' = ysin(y)/(1+x^a) > > lim (for x -> infty) y(x) = b > > Existence and uniqueness of C^1 solutions is the matter. This seems even > harder than the previous. Presumably you want this for x >= 0 (if a is not an integer, x^a is not nice for x < 0). Hint: phase plane and separation of variables. The result will depend on whether the improper integral int_0^infty (1+x^a)^(-1) dx converges or diverges. -- 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 10 Jun 2010 16:26 > Hint: phase plane and separation of variables. The result will depend on > whether the improper integral int_0^infty (1+x^a)^(-1) dx converges or > diverges. By phase plane analysis do you mean phase portrait?
From: Robert Israel on 10 Jun 2010 22:23
On Thu, 10 Jun 2010 22:26:35 +0200, Joubert wrote: >> Hint: phase plane and separation of variables. The result will depend on >> whether the improper integral int_0^infty (1+x^a)^(-1) dx converges or >> diverges. > > By phase plane analysis do you mean phase portrait? Yes. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada |