@Markiyan Hirnyk 

You say "The only closed form solution of the nonlinear ODE under consideration it can find is y(x)=0", but if you read the help page (?dsolve or ?dsolve,details) you see that dsolve is expected to always return a general solution , or otherwise return NULL. So this solution y(x) = 0 is not according to what is written in the help page, and so it is wrong even if mathematically a correct particular solution.

The actual problem was in one of the internal routines for computing solutions to nonlinear 2nd order ODEs in terms of elliptic JacobiSN functions, a condition being tested required an additional simplification, it is fixed now, the fix is available as usual in the Maplesoft R&D webpage for Differential Equations and Mathematical Functions.

Besides that, you could see that dsolve can compute a general solution for this equation if you try the Lie symmetries option, as in

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Hi

I'm rather busy in this moment for several weeks ahead;let's see if I can give this one a look carefully and return useful feedback sooner than that.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@Michael_Watson 

Maple_Question_7.8.14_(reviewed).mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@Alejandro Jakubi 

Just to clarify: the R&D udpates for Physics, Differential equations and Mathematical functions, are official Maplesoft updates. These  R&D updates pass through all the related Maplesoft internal test suites before being posted in the R&D pages for download. These R&D updates are however different from the more general dot updates in that R&D ones have not passed through beta testing as the dot updates do, and for that reason the R&D updates mechanism is so much more agile. In Physics, for instance, we frequently have one or more updates per day.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@oldstudent 

I am not sure about your question: if you refer to the "DEs and Mathematical Functions Updates" mentioned in the title of this post, this update is already available for download for everybody in the Maplesoft R&D Differential Equations and Mathematical Functions webpage. This update, together with the update of the Maple Physics package are currently distributed only as a download from these two webpages, so you do not get them via tools -> check for updates.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@Alejandro Jakubi 

The status is: work in progress. I preferred to start where the gaps were bigger, clearly the 42 elliptic functions. Not what you asked but anyway some details: these "elliptic functions" include the 13 JacobiPQ, 13 InverseJacobiPQ, JacobiZeta, the 4 JacobiTheta, the 4 Weierstrass, and then the more familiar F, E, E', K, K', Pi and Pi' - 7 functions for which, in addition, we have an issue with their definition implemented in Maple in the 90's: it is non-standard, making things more difficult, because there is basically no literature to consult that uses these definitions.

There is always the issue of 'backwards compatibility" to consider but, generally speaking, the natural thing would be to go ahead redefining these F, E, E', K, K', Pi and Pi' elliptic functions according to the literature, resolve any differences by always following the modern and thorough NIST Digital Library of Mathematical Functions project, and finish with this historical issue in one go. The NIST project is the XXI century continuation of the work by Abramowitz & Stegun, Gradshteyn & Ryzhik, Bateman, Byrd & Friedman, P.B.M, etc..

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@Alejandro Jakubi 

Thanks; it's fixed now. Regards.

Edgardo

@oldstudent 
I added below an incomplete list of Maple strengths that in my opinion disproves your conclusion.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@USPAS2014 
Just about your comment on availability: the mini-course is linked in the Maplesoft R&D Physics webpage. - last link in the column on the right.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@Alejandro Jakubi 

Indeed I do have it my .mapleinit. About having this as a new command: generally speaking, when certain functionality can be used through options we will not add it as a new command. This case however is one where the combination of arguments is cumbersome enough and the functionality is used frequently, kinda justifying an exception. I forwarded now your suggestion to the people who take care of plots.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@rashmi 

I understand you can perform this computation in current Maple 18, but am not sure if the DifferentialGeometry package in Maple 12 could help you for that purpose, probably yes, give it a try. The starting point would be the help page ?DifferentialGeometry,OverviewOfGeneralRelativity.

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@trace 

I will have time to give a look at your paper by Friday, hopefully before that. By the way nice worksheet the one posted by Torre using the DifferentialGeometry package for this problem.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@trace 

I am not sure I understood your worksheet as you intended, here are two possible interpretations. In the first one you seem to want to set the metric as your expression (2)

> (dt+4*w*(1/m^2)*(sinh((1/2)*(m*r))^2)*dphi)^2-sinh(m*r)^2*(1/m^2)*(dphi^2)-dr^2-dz^2

and then compute the value of Ricci[2,2]. If this is all what you want, and assuming that the coordinates are already set, just enter

> Setup(metric = %)

And the value of Ricci[2,2] is then given by

> Ricci[2,2]

And that is all. 

If however your question is about how to perform a transformation to then compute Ricci[2,2], suppose then you want to perform a transformation on a tensorial expression, and by that I mean you have the set of transforming equations, say tr, with the old variables on the left-hand sides and the new variables on the right-hand sides, or the other way around (new = old instead of old = new). Suppose now that you want to compute the components of a tensorial expression in the new coordinates, for example: the component [2,2] of the Ricci tensor. This is the sequence of steps:

1) transform coordinates on the spacetime metric

> TransformCoordinates(tr, g_[mu,nu]);

The output will show up in matricial form. If you prefer to see it as the square of the line element, then pass also the optional argument 'output = line_element'. Either way, if the output is according to what you expect, then:

2) set this result as the new value of the metric

> Setup(metric = %);

And that is all. You can now compute the value of any tensor directly, for example, as in

> Ricci[2,2]

The relevant thing then is the transformation tr. Reading your worksheet I have the impression you meant to use a transformation that can be represented in matricial form by

> M := Matrix([[1, 0, 0, 0], [0, sin(m*r)*(1/m), 0, 0], [0, 0, 1, 0], [0, 4*w*(sinh((1/2)*(m*r))^2)*(1/m^2), 0, 1]])

Am I understanding your worksheet correctly? Note also that the ordering of coordinates I am using is [r, phi, z, t], so t is the 4th, and hence also the 0th coordinate - this is the convention in the Physics package: all 0th components are entered as 4th components and you can refer to them using 0 or 4, as you prefer.

I noticed also that the first two lines are not actually equal: simplify((1)-(2)) does not give 0 - I am not sure whether that is consistent with what you present).

In summary: I suggest you to write the transformation explicitly and then use TransformCoordinates. Please feel free to ask again if you cross with obstacles; I'd be interested in seeing your development.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

@trace 

Thanks for posting your worksheet mm.mw; the answer appears further below

Edgardo S. Cheb-Terrab 
Physics, Differential Equations and Mathematical Functions, Maplesoft

@trace @escorpsy

Download mmm_(reviewed).mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft