According to Wikipedia, Abel's first order odes have general analytical solutions, due to "Panayotounakos, Dimitrios E.; Zarmpoutis, Theodoros I. (2011)" where the claim is that, if I understand it right, all Abel ode's can be solved analytically.

https://en.wikipedia.org/wiki/Abel_equation_of_the_first_kind

"Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)"

Maple is very good on solving ODE's, but some Abel ode's it can not solve. For example

restart;
ode:=diff(y(t),t)= y(t)^3+exp(-5*t);#_Abel
ode:=diff(y(x),x) = (1+x^3*y(x))*y(x)^2;#_Abel
ode:=diff(y(x),x) = y(x)^2-a*x*(1-x^(n-1))*y(x)^3;#_Abel
ode:=diff(y(x),x) = a*y(x)^2+x*y(x)^3*(b+c*x^(n-1));#_Abel
ode:=diff(y(x),x) = f0(x)+f1(x)*y(x)+f2(x)*y(x)^2+f3(x)*y(x)^3;#_Abel
ode:=(tan(x)*sec(x)-2*y(x))*diff(y(x),x)+sec(x)*(1+2*y(x)*sin(x)) = 0;#_Abel, `2nd type`, `class A`
ode:=x*(a+y(x))*diff(y(x),x)+b*x+c*y(x) = 0; #[_Abel, `2nd type`, `class B`]
ode:=(g0(x)+y(x)*g1(x))*diff(y(x),x) = f0(x)+f1(x)*y(x)+f2(x)*y(x)^2+f3(x)*y(x)^3;#_Abel, `2nd type`, `class C`

etc..

All the above Abel ode's are from Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960

I am just asking what is the status of this. Is it true there is now a method to solve all these exactly and may be Maple's implementation is not in yet for this? 

btw, I found the description of solution methods in https://fr.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Abel much easier to follow to learn how to solve Abel ode's. That paper mentioned on Wikipeida, I had hard time following after the 3rd page. (need more time to study it).

Could someone please comment on the status of solving Abel's first order ODE's in Maple and if it possible now to solve them all analytically?

Why when trying to substitute a term in denominator, subs does not work, when this term is product. But it works when this term is single variable?

subs((x*y)=t,1/(x*y));

does not work. i.e. it does not return 1/t

But this works

subs(z=t,1/z);

and returns 1/t

algsubs does not work either on the first example above. 

Just wondering why, that is all.

I do not think Maple help on the subject is easy to follow as the exampels are not clear. I was hoping someone can give a very simple example.  

Now, I run long script, over say 1,000 problems. This takes long time each time I run it. Since each problem is completely independent of others, and there is no shared data at all (for each problem, its output is written to separate file), I am thinking of using the parallel programming in Maple, and hoping this will speed it up. I have an intel PC, with modern CPU. I think it has may be 10 or 16 cores, not sure now, so in theory it should be faster to complete.

For a very basic example, suppose I have this sequential program

foo:=proc(i::integer)
   return i^2;
end proc;

for i from 1 to 10 do
   foo(i);
od;

Where foo() is now called 10 times, one after the other. How would one change the above to make it run in parallel?

I know I need to call Threads:-Task:-Start and then use Threads:-Task:-Continue 

But I am not sure how to use Threads:-Task:-Continue to tell it to call foo in parallel passing each task 1,2,3,.... in turn, and wait until they are all done. This is the part not clear to me how to do even after looking at the help example under Task Programming Model

Could someone show how to do the above for this simple example?   

Since I have 1,000 problems, do I need to create 1,000 tasks at once, one for each problem and wait for them to all be done? what if I have 10,000 problems, creating 10,000 tasks at once might not work, would it? or will Maple handle this internally by queuing the creation of tasks as needed? This part is also not clear to me.

thanks

Maple seems to suffer from too many levels of recursion issues. Here is another just found

restart;
current_solution:=y(x)-3/2 = 1/(x+2)^4*(-(x+2)^5*RootOf(-2+(x^5*C[1]^5+10*x^4*C[1]^5+40*x^3*C[1]^5+80*x^2*C[1]^5+80*x*C[1]^5+32*C[1]^5)*_Z^25+(5*x^5*C[1]^5+50*x^4*C[1]^5+200*x^3*C[1]^5+400*x^2*C[1]^5+400*x*C[1]^5+160*C[1]^5)*_Z^20)^20*C[1]^5+1)/RootOf(-2+(x^5*C[1]^5+10*x^4*C[1]^5+40*x^3*C[1]^5+80*x^2*C[1]^5+80*x*C[1]^5+32*C[1]^5)*_Z^25+(5*x^5*C[1]^5+50*x^4*C[1]^5+200*x^3*C[1]^5+400*x^2*C[1]^5+400*x*C[1]^5+160*C[1]^5)*_Z^20)^20/C[1]^5;

candidate_sol := limit(current_solution,C[1] = infinity);

Error, (in depends) too many levels of recursion

The problem with these is that they can not be cought by a try/catch. So the whole program/script comes to halt since error can not be cought.

Similar ones I found can be found here

https://www.mapleprimes.com/questions/229988-Error-in-Toolsmap-Too-Many-Levels
Error, (in tools/map) too many levels of recursion

https://www.mapleprimes.com/questions/229872-Error-in-Discontzero-Too-Many-Levels
Error, (in discont/zero) too many levels of recursion

https://www.mapleprimes.com/questions/229951-Random-Error-Error-in-EngineDispatch
Error, (in Engine:-Dispatch) too many levels of recursion

I am  not sure why these happen too often. Maple 2020.1 on windows 10. Physics version 724

restart;
expr:=a^2*(2*a^2*p^3-a^2*((p^2+1)^2*(a^2-1))^(1/2)-((p^2+1)^2*(a^2-1))^(1/2)*p^2+2*p*a^2-2*p^3+((p^2+1)^2*(a^2-1))^(1/2)-2*p)/((p^2+1)^2*(a^2-1))^(1/2)/(p^3-((p^2+1)^2*(a^2-1))^(1/2)+p)/(a^2-p^2-1);
int(expr,p)

Gives

why does Maple give division by zero?

Here is the result from integration package on Mathematica

ClearAll[a, p];
expr = a^2*(2*a^2*p^3 - 
       a^2*((p^2 + 1)^2*(a^2 - 1))^(1/2) - ((p^2 + 1)^2*(a^2 - 1))^(1/
           2)*p^2 + 2*p*a^2 - 2*p^3 + ((p^2 + 1)^2*(a^2 - 1))^(1/2) - 
       2*p)/((p^2 + 1)^2*(a^2 - 1))^(1/
        2)/(p^3 - ((p^2 + 1)^2*(a^2 - 1))^(1/2) + p)/(a^2 - p^2 - 1)
<< Rubi`
Int[expr, p]

Which it can integrate. Result is a little long. (removed since looks too long)

But my question really is not why Maple could not integrate it, but why the division by zero? 

Maple 2020.1