@Kitonum 

Indeed, the integral can be computed. I updated the answer taking that into account, and added comments regarding why the explicit solution with y(x) isolated is valid only in some region.

It would be useful is you:
a) explicitly show the solution you are expecting, as in "A = B" where A is each of the unknowns of your system and B is their form such that the PDE system cancels out.
b) perhaps more importantly, this system you show looks like the system for the infinitesimals of a PDE problem. If so, it would be useful if you could include in your worksheet the PDE problem and how you are computing the linear PDE system for its infinitesimals.

With a) and if it is also the case b) all explicit, it is possible to provide the feedback you are asking for.

Best wishes for 2023!

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

@C_R 
What went wrong is that you input those lines in 2D-Math notation, and then "ⅆ" became "d", which is not what we want. 

Download Differential_format_in_integrals_with_tweak_(reviewed).mw

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

@C_R 
Indeed, as @acer says, I saved the mw using a version of Maple not yet released. It is fixed now, the mw attached to the post is saved with Maple 2022.2 and can be executed without interruptions.

Best and the best wishes for 2023!

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

Hi @Jamie128 
End of the year, slowing down; I'd suggest you to post a worksheet with the modified gravity Lagrangian and what you'd expect as field equations; if possible, show the steps - as a clear sequence, within the text, or point somewhere where we can see the steps you have in mind, and so, what you expect - and from there we see: either that is possible right away, or it might require an adjustment (evolution) of the code.

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

@zenterix 
_i is, indeed, something special after you load the Physics:-Vectors package, e.g. via with(Physics:-Vectors). After that, _i is a unit vector in the x direction, with its mathematical properties all active (see the help page ?Physics:-Vectors)

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

In order to give you feedback it is necessary to reproduce the problem - not possible with your description. Could you please upload a worksheet showing the problem (include text pointing at the problem and what is what you were expecting please). For that purpose, use the Green arrow you see when you write a question or reply to it. Thanks

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

@C_R 
Thanks for your comments. The equation returned is " expression = 0 ", which is mathematically the same as " - expression = 0 ". In particular, I tweaked the design so that if Lagrange's equation is of the form "- A - B = 0", then multiply by -1 to return "A + B = 0". I will give this tweak further thought.

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

@Scot Gould 
Indeed. Count on that.

Meantime, regarding the problem in PackageTools mentioned in this post, I added a fix to PackageTools within the Maplesoft Physics Updates v.1357, so after installing that version of the Physics Updates using the MapleCloud toolbar (that works), you can install subsequent versions as usual by entering Physics:-Version(latest) directly in the GUI (worksheet or document).

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

@Rouben Rostamian  

The problem is still that you don't have the current version of the updated code - Maplesoft Physics Updates. Here are the details:

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

Hi @Rouben Rostamian  
I see your version of the Physics Updates is 1342. I have v.1355, and am unable to reproduce the problem. I get

So, could you please update to the current Updates; or otherwise execute tracelast after the error message appears (you may need to repeat the operation twice). That may turn ON the lights. Without reproducing the problem, it cannot be "fixed".

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

@nm 
Sigh ... you are right. Somehow, Maple 2022.2 has this problem. You can still install the Updates clicking the MapleCloud toolbar. I tried now, it works.

@nm 

Are you trying from TTY Maple? If so, that is the problem, reported already. You need to update from Maple's GUI, the worksheet, as in open a Maple worksheet and there input Physics:-Version(latest), or use the icons of the MapleCloud toolbar. Both methods work fine for everybody as far as I know.

@acer 
Very interesting cocktail of commands ... Have you tried with different orderings for the output of indets(expr,suffixed(P)? That, I think, is the important question, since the output of indets is a set and the ordering is not predictable with ease. 

Either way, behind the scenes, it is `simplify/size` doing key parts of the job (if you assign `simplify/size` := u -> u; the superb compactification you achieved in your last input line disappears entirely). But I wonder whether the core idea I see in thaw(simplify(collect(expr, [G1, P2, P5], freeze@simplify))) - if it is not so sensitive to the ordering returned by indets (I see you are choosing by hand [G1, P2, P5]) - could be somehow used to improve `simplify/size` itself. In the latest Maplesoft Physics Updates, there is a version of `simplify/size` that works better than the one in Maple 2022.2, but it doesn't achieve the compactification you achieved. The codegen[optimize])(..., tryhard) mentioned by @Carl Love is also an excellent suggestion. This command, `simplify/size`, is relatively good, but being key functionality, any improvement is very welcome!

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

@Ahmed111 
After removing that unnecessary transformation equation, dchange does not interrupt with an error message anymore; that answers your question. Fix also pde -> pde1 in your input and check all of it. Your transformation - the way you present it - involves var[1] and var[2]. There is however nothing like that in the equation you are transforming. Therefore, dchange "does noting" - because nothing is to be done. Your old variables are x and t, only. See the help page ?dchange for examples on how to use the command.