@ecterrab 

Thanks. I knew about your papers and I've just started studying the first one. But I have a small question if I may.

In the Maple help,

https://fr.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Abel 

It says if Abel ODE has f2=0, and the Abel invariant do not depend on x , then the ODE can be solved directly.

But how directly? Since it is still Abel ODE. For example, Kamke #38 has f2=0 and constant invariant:

ODE:=-a*y(x)^3-b/x^(3/2)+diff(y(x),x)=0; #kamke 38

Which has, by comparing the above to y'=f0+f1*y+f2*y^2+f3*y^3 the following

f0:=b/x^(3/2);
f1:=0;
f2:=0;
f3:=a;

DEtools:-odeadvisor(ODE);

[[_homogeneous, `class G`], _rational, _Abel]

And so the Abel invariant is 

f0:=b/x^(3/2); 
f1:=0; 
f2:=0; 
f3:=a; 
inv:=-(-diff(f0, x)*f3 + f0*diff(f3, x) + 3*f0*f3*f1)^3/(27*f3^4*f0^5)

So we see the invariant does not depend on x. But now what? How to solve the ODE "directly". This is the part I am missing. Since it is the same ODE, i.e. Abel. Nothing changed.  

Is there some additional transformation needed? The transformation mentioned in the help, is when f2<>0, in order to eliminate f2. But this is not the case here.

What is it I am missing here? 

thanks

@Joe Riel 

Ok, So what is the internal representation of 1/z ? would that not be z^(-1)? If so, then why subs(z=t,1/z); worked, since there is no z term in the expression, there is only z^(-1)?

What would you recommend then to replace (x*y) in the expression 1/(x*y) with some other value such as t which would also work if (x*y) was in the numerator or in the denominator?

@Joe Riel 

"Something I'd like to see is object inheritance."

Maplesoft seems to have limited resources and is a small company compared to the competition.

Therefore It would be much better if it uses these resources on things that are much more useful  and practical than adding inheritance and multiple inheritance, which might end up being used by very few people to do CAS programming. 

Improving the debugger for example, will be much much more useful to many more people as well as improving the Latex export. These things are used by many people all the time.

Maplesoft should also concentrate its resources more on core Mathematics and less of fancy features trying to follow the competition.

I'd rather see Maplesoft use its limited and important resources to make Maple solve more PDE's, ODE's, equations and integrals for example. (in addition to improving the debugger and Latex). 

I have not even mentioned the robustness of the software. Maple hangs on me each day may be 2-3 times whenever I run a long computational script, and at random locations, due to memory issues it seems in mserver.exe.

It seems to me that doing stress tests on Maple to fix such problems is much more important use of your resources.

@Carl Love 

yes, great answer as usual. I've been busy and now got to try it and it works very well. thanks.

could these be build-into Maple, as an extra option to plot command, called 'plot_theme' or such? 

Mathematica has this option, they call it `PlotTheme` https://reference.wolfram.com/language/ref/PlotTheme.html  and one can select many themes. 

It will be good to have such functionality as part of Maple's plot command as well.

@Carl Love 

May be we are using different Maple versions?
 

Download check_3.mw

@ecterrab 

Thank you very much for this nice hint. I had no idea one could do this in Maple. This is very useful and I will use it more now.

I think this ODE comes from the book "Ordinary differential equations and their solutions. By George Moseley Murphy. 1960". 

Actually the solution I got, before simplifying it is very close to Maple's

I used Maple to help with integration and simplification of expressions during solving it.

thanks again. Attached PDF if needed.

foo3.pdf

@vv 

opps, sorry about that. I was not trying to confuse any one. I confused myself, since I copied the data into a Matrix() and did not notice that <<|>> notation and Matrix([]) do not geneate the same matrix, but the transpose of it. That is why.

Thanks for pointing that out. 

I do not like to use << | >> notation in Maple, and prefer to use Matrix( [] ), that is all.

@rameen hamood 

are you sure you copied it right? it says

let A=<< 87, -66, -90, 12, 48>|<-36, -40, -82, -54, 15>|<99, 79, 76, -31, 27>|<-69, 15, -10, 45, -9>>

and x= <9,-4,2,-17>.

But A is 4 by 5, and x is (if you look at it as column vector) is 4 by 1. So inner dimensions do not match. Even if you look at x as row vector, it is 1 by 4, and inner dimensions do not match either.

But if you view x as matrix, then x.A works, like this

restart;
A:=Matrix([ [87, -66, -90, 12, 48],[-36, -40, -82, -54, 15],[99, 79, 76, -31, 27],[-69, 15, -10, 45, -9]]);
x:=Matrix([9,-4,2,-17]);
TA:=x.A

which gives

         [2298, -531, -160, -503, 579]

Do not know if this is what being asked.

is what you asking how to multiply a matrix by a vector in Maple?

@Kitonum 

would you know why Maple choose in this example to give the solution the way it did, instead of the normal way, which would be 

                      y(x)=arccos(ln(x)+1)*x, y(x)=-arccos(ln(x)+1)*x 

i.e. a sequence of 2 solutions.

I find it very strange, why out of a sudden Maple choose this new new way to write out the solution. I find it confusing way to write out the solutions, given that Maple typically writes out solutions in clear way.

@vv 

I really understand what you are saying, I guess this is a philosophical question in a way.

Just as I view C=A/B the same as -A+C*B=0 just written differently, So I viewed   dy/dx=A/B the same as  -A dx+ B dy=0 just written differently and now it is exact, but if one views it as -A/B dx + dy=0 it is not exact.

So it depends how it is written.

But I ended up checking for both cases now. If the first form fails test for exact, then I try the second form of the ODE. So I am all set.

But I do get your point. It would hard for Maple to check different forms each time to find out.

@vv 

I can't enter Latex here. Here is screen shot

It is the same ODE, just written differently?  You are viewing moving the denominator to the other side as multiplication by integration factor? I did not look at it this way. I just viewed it as same ODE just written different.

@ecterrab 

thanks. I remember now running into this some time ago. The problem is that odetest worked using C[1] in my solutions may be for 99.99% of the time and only very rarely it failed. That is why I thought I did not need to change my code to use _C1 instead of C[1]. 

I do not use Maple's own solutions in odetest, but my own solutions. 

But it is resolved now. I simply changed my code to use _C1 and _C2 instead. I was first worried about using _C1 since I read that user code is not supposed to use variables with _ in front of them, due to possible conflict with Maple's own internal variables.

Issue is solved. Sorry if this caused any problems.

@AHSAN 

this is not what you wrote in your original question. In addition, I no longer understand what you are writing. In (22) you say diff(p(x),x) and in (23) you write diff(p(y),y)=0.  

Are these pde's and not ode's? I'll let someone else try to help you, as I am not able to understand your problem. good luck.