gerbil Posted April 15, 2012 Share Posted April 15, 2012 Hello All, Can anyone find an analytic solution (preferred, not implicit) to the following equation: \dot{x} = -alpha \sqrt{x} + \xi(t) x(0) > 0, t \in [0, T] and T such that, x(t) > 0. \alpha > 0 and \xi(t) is any smooth function. A solution for a any specific non-trivial \xi(t) is also appreciated. Best, Miki Link to comment Share on other sites More sharing options...
Xittenn Posted April 15, 2012 Share Posted April 15, 2012 (edited) Hello All, Can anyone find an analytic solution (preferred, not implicit) to the following equation: [math] \dot{x} = -\alpha \sqrt{x} + \xi(t) [/math] [math] x(0) > 0 [/math], [math]t \in [0, T][/math] and [math]T[/math] such that, [math]x(t) > 0[/math]. [math]\alpha > 0[/math] and [math]\xi(t)[/math] is any smooth function. A solution for a any specific non-trivial [math]\xi(t)[/math] is also appreciated. Best, Miki For latex use [math] [/math] without the underscore! Edited April 15, 2012 by Xittenn Link to comment Share on other sites More sharing options...
gerbil Posted April 16, 2012 Author Share Posted April 16, 2012 Testing [math] 1, x, \frac{x^2}{2} ... [/math] Link to comment Share on other sites More sharing options...
Bignose Posted April 17, 2012 Share Posted April 17, 2012 Clearly some analytic solution exists for a non-trivial [math]\xi(t)[/math] Take [math]\xi[/math] = a constant (which is clearly smooth). Because that can be turned into one of the forms in the "Integrals With Roots" section: http://integral-table.com/integral-table.html#SECTION00004000000000000000 Link to comment Share on other sites More sharing options...
gerbil Posted April 17, 2012 Author Share Posted April 17, 2012 Clearly some analytic solution exists for a non-trivial [math]\xi(t)[/math] Take [math]\xi[/math] = a constant (which is clearly smooth). Because that can be turned into one of the forms in the "Integrals With Roots" section: http://integral-tabl...000000000000000 Yes. For the case where [math]\xi(t)=\mathrm{Const}[\math] there exist an analytic solution. However, the solution is implicit. How about solutions to the case where [math]\xi(t)[\math] is a polynom. .. Yes. For the case where [math]\xi(t)=\mathrm{Const}[/math] there exist an analytic solution. However, the solution is implicit. How about solutions to the case where [math]\xi(t)[/math] is a polynom. .. Link to comment Share on other sites More sharing options...
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