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. ..

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

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

Just for my reading purposes!

 

On a scale of 1 - 10 in terms of analytically solving first order nonlinear ordinary differential equations how hard is this?

I dont think that the word "hard" is relevant here. The problem is that I'm not sure that my equation can be solved analytically (and I'm not sure about the opposite either).

Maybe there exists a tricky substitution that solves this equation and maybe there isn't. It is a bummer ...

But there are some cases in which nonlinear (first-order) equations can be solved, for example, the Bernoullis' equations.

Your parentheses don't match in 1/((Sqrt(x))*(atan(x)), but given your u-substitution, I gather you are asking about [math]\int \frac 1{\sqrt x\,\arctan x}\,dx[/math]

 

Your integrand is a transcendental function. There is no analytic solution in the elementary functions to this integral.

Yes. That's what I was afraid of. Well, I'll just post the original ODE and maybe you'll have an idea how to solve this one, before I give up.

$\dot{x} = - \alpha {x}^{1/2} \arctan{(kx)}$

for $x > 0$ and $t \in [0, T]$ , where $T < \infty$ and $k > 0$.

Best,

Miki

Hello all,

I'm trying to solve analytically a differential equation and I encountered the following integral,

Integrate [1/((Sqrt(x))*(atan(x)), dx]

The change of variable u = sqrt(x) brings this integral to,

2 * Integrate [1 / atan(u^2), du]

Now what ... ?

Any help will be highly appreciated.

P.S.

I attached a PDF file with the problematic integral.

Question.pdf

Hello All,

I have a question related to the fascinating topic of "Boundary Layer Theory".

Take for example the simple equation

mx'' + kx' + cx = 0 where k and c are finite constants and (m -> 0), (The derivative sign (') is by t)

x(0) = 0

x'(0) = A > 0.

In order to achieve a comprehension on what's going on near the point t = 0 when (m -> 0), it is suggested to take the change of variables t* = t/m.

By that the former equation becomes,

x'' + kx' + mcx = 0 (Now, the derivative sign (') is by t*). Now, moving into the limit or alternatively noticing that mcx is negligible, we can drop it and we get,

x'' + kx' = 0 and now the solution is fairly easy.

Books call this solution: "The inner solution" of the equation, since this solution describes the solution inside a boundary layer.

My question is as follows:

The idea of taking the change of variables t* = t/m, is just a mathematical trick or does it have a meaning (e.g. streching of the time scale, etc.)?

If it has a meaning, i'd like to know its intuition.

Best,

Miki

Thanks!

Can you refer me to a link / paper or a book that deals with the convergence of exp(A(t)) ?

Thanks,

Miki

Hello All,

My question is on the problem of solving the ODE x' = A(t) x (with

x(0) = x0) and A belongs to R(NXN).

During my research I've encountered the question of whether the sum

exp(A(t)) = sum ( I + A(t) + 0.5*(A(t))^2 + ... ) is uniformly

convergenct with respect to t, where t belongs to [0, T] and T is

finite.

One can assumes smooth matrix A(t) and A(t) is invertible.

Moreover, it can be assumed that A(t) is also skew-symmetry if it

helps, and that the limit exp(A(t)) exist and finite for all t.

The question comes from the problem of differentiating exp(A(t)) term

by term. ...

So, is it true that "sum ( I + A(t) + 0.5*(A(t))^2 + ... ) " is uniformly convergent in t?

Best,

Miki