Non Linear ODE

[gerbil]

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

[Xittenn]

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

[gerbil]

Testing [math] 1, x, \frac{x^2}{2} ... [/math]

[Bignose]

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

[gerbil]

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. ..