@vv 

"For example F(x,y) = x * ( y + x*sqrt(x*y) + sqrt(x^3*y) )
can be split only for x<0, "

Yes, good point. So I guess I have to assume x>0 for all test cases. This is in the context of solving a first order ODE by separation. So suppose the ODE was

               y'(x) = F(x,y)

and F(x,y) was your function above. If one can separate it, then the ODE becomes easy to solve. But most HW problems normally do not say x>0, but I think it is safe to assume this. Unless otherwise stated. I need to worry about this.

btw, your F seems to give dsolve hard time.

@Kitonum 

I implemented this `` trick in what follows. It gets little tricky, since I have to watch out for op(1,op(1... all the times. Does this look ok for you? 

Given sqrt(1-y^2)/x; I want to write is as sqrt(|1-y^2|)/x; and then remove || to obtain the original expression back. Using subsindents. So I wrote

restart;
Y:=sqrt(1-y^2)/x;
Y:=subsindets(Y,anything^({1/2,-1/2}),ee->abs(``(op(1,ee)))^(op(2,ee)));     
Y:=subsindets(Y,abs(anything)^(1/2),ee -> op(1,op(1,op(1,ee)))^op(2,ee));

Which works. I am just getting little dizzy counting ops in op(1,op(1,op(1,ee)))^op(2,ee) to get it right.

I wish there was a simple way for Maple to just leave what is inside abs() the way it was and not muck it up.

@Carl Love 

fyi, I found some problems.

expr:=sqrt(1-y^2)/(2*sqrt(x));
r:=Separate(expr,[x,y]);

It should be  sqrt(1-y^2) and 1/(2*sqrt(x)). And

expr:=sqrt(1-y^2)/x;
r:=Separate(expr,[x,y],method=tryhard);

The result should be   sqrt(1-y^2) and 1/x

And

expr:=sqrt(1-y^2);
r:=Separate(expr,[x,y],method=tryhard);

It should be  1 * sqrt(1-y^2).

I think these all are from same "issue". Having complex numbers pop up like this, makes it hard to use. I will add these cases to the list of cases in my original post now.

I am sure there is a way to simplify the result obtained back to what it should be with assumptions and combine(...,radical). So the output is not wrong, but hard to use as is. A post-processing step is now needed.

thanks

@Carl Love 

very good. Now with tryharder option (I saw this in your code, but did not know I can pass it in), your function can do them all correctly. congratulations.

I did not say that "Maplesoft can achieve with a "build-in" function something better than we can make here " I simply said such a function ought to be build-in Maple. Maplesoft can take your function for example and use it.

There are two important and useful functionality missing now in Maple itself: They are finding additively separable functions 

f(x,y,...) = g1(x) + g2(y) + .....

And product separable functiuons

f(x,y,....) = g1(x) * g2(y) * ....

https://ieeexplore.ieee.org/document/1104018/

Neither Maple nor Mathematica as well, have support for finding such decomposition.

I will test your function now more with other functions I have, and will use tryharder option.

thanks

@Carl Love 

Nice. Your function works well. I only found it fails on the first one in my list of tests I posted above. This was the hardest one.

expr:=arcsin(x)/(sqrt(1-x^2)*y^2);

This is clearly separable. But your method does not seprate it

Separate(expr,[x,y]);

                         FAIL

It should be

Sometimes doing things by CAS is much harder than doing things by hand. By just looking at the expression one can see it is separable. Why can't computers do the same?  :)

what I have I am still testing, but it is based on this paper   https://arxiv.org/ftp/math/papers/0210/0210167.pdf

by C. P. Viazminsky. The method is basically covered in equations (2.1) to equation (2.8).

This is something that Maple itself should have as build-in function. This will be very useful to have in Maple.  May be someone inside Maplesoft can look into adding such a function to Maple.

@Kitonum 

thanks for all the answers. But I found a problem

Your method works for ln(x) when ln(x) is both in numerator and denominator.

But for sqrt(x), which I want to change to sqrt(abs(x)), it only works when sqrt() is in the numerator.  

expr:=sqrt(x);
applyrule(sqrt(t::anything)=sqrt(abs(t)), expr);

is OK. But not for:

expr:=1/sqrt(x);
applyrule(sqrt(t::anything)=sqrt(abs(t)), expr);

does not work. Since I do not know where the subexpresion will show up in the expression, this will be a problem. I do not know why applyrule works for both 1/ln(x) and ln(x), but only for sqrt(x) and not for 1/sqrt(x).

@vv 

Thanks. This looks better. But using these arbitray "a" and "b" makes it little hard to use.

I wrote a a Maple function based on paper "On Separation of Variables by C. P. Viazminsky". I am still debugging and testing it. But this is what I get using it. on the test functions I posted above

can not split arcsin(x)/(-x^2+1)^(1/2)/y^2
x*y = (x) * (y)
x^2*y+x^2 = (x^2) * (y+1)
2*x+1 = (2*x+1) * (1)
cos(2*x) = (cos(2*x)) * (1)
y = (1) * (y)
(x-1)/y = (x-1) * (1/y)
ln(y^2+1) = (1) * (ln(y^2+1))
can not split -x*y^2+y^2-x+1
can not split x^3*y+x^2
can not split 1/(x+y)
(1+x^(1/2))/(1+y^(1/2)) = (1+x^(1/2)) * (1/(1+y^(1/2)))
(-x^2*y^2-x^2+y^2+1)/x^2 = ((x^2-1)/x^2) * (-y^2-1)
3*x^2*y^2+2*x*y^2 = (3*x^2+2*x) * (y^2)
can not split 3*exp(2*x)+2*y
can not split (5*x^(1/2)-y)/x
can not split 2*x*y+3*x^2*exp(x^2)
can not split (y^2+x+1)^(1/2)
1 = (1) * (1)
can not split x*y+1
4*x^3*y-y = (4*x^3-1) * (y)
can not split exp(x^2)+2*x*y
can not split -(y-exp(x))/(x-2)
arcsin(x)/(-x^2+1)/y^2 = (arcsin(x)/(x^2-1)) * (-1/y^2)
can not split -(y-exp(x))/(x-2)
4*x^3*y-y = (4*x^3-1) * (y)
can not split (x^2*y+y)*y^3*exp(-x-y+1)*3^(-x-y)*(x^2*y-y)^(1/2)

The only problem is the last one. Based on the function is_separable(f,x,y) that I posted above, here is it again

is_separable:=proc(f,x,y)
  #checks if f(x,y) is separable to X(x)*Y(y) or not
  #but does not do the separation. The separation is done by
  #another function I have
  if diff(simplify(diff( simplify(ln(f)),x)),y) = 0 then
     return(true);
  else
     return(false);
  fi;
end proc:

The last function is supposed to be seperable.

is_separable((x^2*y+y)*y^3*exp(-x-y+1)*3^(-x-y)*sqrt(x^2*y-y),x,y);

true

Yet, the function I wrote does not separate it. my function also does not separate the first one, which is sqrt(x-1)*sqrt(x+1)/(sqrt(-x^2+1)*y^2); but your SplitX does. But since your use all these extra constants, it makes it hard for me to use now.

Thanks for all the help.

@John Fredsted 

hi. Your method also has same issue as I just mentioned below.

expr:=-y+4*x^3*y;
s,r := selectremove(z -> has(z,x) and has(z,y),expr):
``(simplify(select(has,r*expand(s),x)))*
``(simplify(remove(has,r*expand(s),x)));

It should be  y*(4x^3-1)

And

expr:=exp(x^2)+2*x*y;
s,r := selectremove(z -> has(z,x) and has(z,y),expr):
``(simplify(select(has,r*expand(s),x)))*
``(simplify(remove(has,r*expand(s),x)));

Which is also wrong.

Hard problem!

@Kitonum 

I tried your method, but unfortunately, it gives wrong results on number of cases. I tried both Splitting and Splitting1. Actually all methods shown in all the answers, do not all work on all cases. This is not an easy problem as it looks.

expr:=-y+4*x^3*y;
result := Splitting(expr,x,y);

It should be y*(4 x^3-1)

And

expr:=exp(x^2)+2*x*y;
r:=Splitting1(expr); 

The above can't be split.

@Carl Love 

" The syntax A:-B searches module A's exports for something named B. On the other hand, the B without a module prefix is just accessing a local "

Ah, but for newbies, it is not clear that A:-B means to search for only the exports of A. I just thought it mean just a naming convention to make it more clear which B is it, weather it is exported or not.  Also did not know if using B with prefix implies local

Good to know. I only been using modules for few days.

But this brings another follow up question.

What if parent has a local proc named foo(), and the child also has a local proc named foo(). Then how will the child call the parent local foo() proc if it wants (in addition to calling its own foo() when it wants) if one is not allowed to use the syntax parent:-foo() to tell Maple which foo() they want to call and without being forced to export the parent foo()? Here is an example

parent_module:=module()
 
 local child_module;
 local foo;

 export parent_entry;

 child_module:= module()
    export child_entry; 
    local foo;

    child_entry :=proc()
     foo(); #How to call the parent foo()??
    end proc;

    foo :=proc()
     print("in child foo");
    end proc;

  end module;

 #local
 foo :=proc()
   print("insider parent foo");
 end proc;

 #public
 parent_entry :=proc()
   child_module:-child_entry();
 end proc;
end module;

I can make the above a seperate question if needed.

thanks

@Preben Alsholm 

Thanks for both answers, Both works.

But I do not like to put the absolute path of child packages inside the $include, since I have to change this later if I move the folder somewhere else or if I want to send this package to someone to use?

So now I put all .mpl files in same folder and use relative path and add $include "child_package.mpl". 

But before loading the main package, I have to set the path, which is OK, since this is done outside the package.

@one man 

on my Maple nops gives an error. How come it worked for you? Which version do you have?

I am using Maple 2018. 

thanks.

@Carl Love 

I have not touched or changed anything. It was OK earlier today. I could do computation. It is only the menu items and help that was causing this error.

It seems something was causing it. The good news, is that I just rebooted the whole PC and now this error is gone !  I can start help OK now.

Something was causing it which I do not know what. But rebooting the PC fixed it.

I could delete this question in this case, or leave if it is of any it could happen for someone else or if it happens again for me. This is first time I've seen this problem with Maple 2018

@acer 

I think I either found a bug in the above function F, or I do not understand what it does when the input is not separable.

I gave it an expression which is not separable. It returns product of two expressions. But this is not separable.   How is one supposed to know if the returned value is the separation/split or may be it it same as the input written in different way? Here is an example

expr := -(y-exp(x))/(x-2);
F (expr, [x,y] );

I also found an expression which it can't separate

expr := 1-x+y^2-x*y^2;
#this can be split to (1-x)*(1+y^2)
F( expr, [x,y] );

And this one

expr:=(1-x^2+y^2-x^2*y^2)/(x^2);
#this can be split to 
fx:=(1-x^2)/x^2;
fy:=(1+y^2);
simplify(expr-(fx*fy));
F( expr, [x,y] );

This is not an easy problem!. I think Maple should have build-in function to do this.

I do not understand your code too well. Too advanced for me to change it. It will help if your function will return a list of the 2 split product, when it can separate them, or 2 empty lists if it can't separate them. This way a user would more easily check, like this

fx,fy := F(expr, [x,y]);

if fx={} or fy={} then
   print("can't separate them");
else
   print("separated them into two functions", fx, fy);
fi;

Thanks

@acer 

I used LibraryTools:-Save as you recommend. I was just asking if there was a different example you refered to. The whole documenation is really confusing in Maple. They do not spell things out very well for newbies. 

For example., in LibraryTools:-Save('name'), it does not say that it will save the contant of "name.mpl" to "name.mla" by default. I found this by trial and error. 

I ended doing this

restart;
read "my_package.mpl"; 
        #note that now Maple currentdir() is automatically 
        #set to where the worksheet is located. This seems to be
        #new in Maple 2018, which is good change.

my_package_name := cat( currentdir(),kernelopts(dirsep),"my_package.mla");

LibraryTools:-Create(my_package_name); 
          #this will create file my_package.mla 
          #in same folder as this worksheet

LibraryTools:-Save('my_package'); 
             #this will save content of "my_package.mpl" 
             #in "my_package.mla" file. This step was a 
             #little confusing. As name passed refers to
             #both the input (.mpl) and the (.mla) with same 

restart;
with(my_package);   #now this will work

#to update the package, after making changes to .mpl, do
restart;
read "my_package.mpl";
LibraryTools:-Save('my_package');

restart;
with(my_package);   #now this load the updated package