@Markiyan Hirnyk 

Hi Markiyan, y=0 is not a general solution, because it has not the necessary number of arbitrary constants. Therefore is not the solution expected from dsolve according to its help page. The fact that dsolve, in addition, returns essential singular solutions doesn't change that fact. By the way y = 0 is also not an essential singular solution of the ode posted, nor a singular solution of any kind, but just a particular solution that can be obtained taking one of the integration constants equal to 0 in the general solution.

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

@Mac Dude 
I am sorry but the update for Maple 18 does not work in previous releases. 

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

@Markiyan Hirnyk 

Look closer into ?dsolve,details. In the Section entitled "Arbitrary Constants, General, Particular and Singular ODE solutions", you read: "By default, dsolve computes the general and essentially singular solutions of a nonlinear ODE; the latter are the singular solutions that cannot be obtained from the general solution by specializing the integration constants in any way."

In the same section, you read: "A general solution to an ODE depends on as many arbitrary constants as the differential order, say N. Conversely, any solution depending on N arbitrary constants is a general solution. Depending on N constants here means there is no possible redefinition of the _Cn in terms of other N independent constants that results in a form of the solution with fewer than N constants."

I suppose you missed this section when reading the page, but there is no possible misunderstanding: a solution of the form y(x) = 0, without arbitrary constants for a 2nd order ODE does not match these statements quoted from the help page ?dsolve,details.

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

@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