These two expressions are the same

e1:=-sqrt(-(exp(-2 + 2*x) - 2)*exp(-2 + 2*x))/(exp(-2 + 2*x) - 2);
e2:=1/sqrt(2*exp(-2*x)*exp(2) - 1);

Is there an automated way to simplify e1 to e2? Below are my attempts. The closest I got is 

simplify(e1) assuming real;

But that still does not give same as e2. I can do it by "hand" as shown. But I like to find automated way since this is done in code without looking at expression. So I can't use the "hand" method there.

We can assume everything in real domain.

Download simplification_june_6_2024.mw

We see that this ode (x + y(x))*(1+diff(y(x),x)) = 0  has 2 solutions, One when (x + y(x))=0 and one when (1+diff(y(x),x))=0. Maple gives 3 solutions. They are correct but why?

Also when changing (1+diff(y(x),x)) to (a+diff(y(x),x)) now it gives only two solution.

Why does this happen? Should it not just return 2 solutions in both cases? and more strange one

(x + y(x))^2 *(1+diff(y(x), x))=0; now it gives 4 solutions. But this is no different. We also have 2 solutions. One when (x + y(x))=0 and one when (1+diff(y(x), x))=0. This time the extra two solutions are complex. 

Download why_extra_solution_from_dsolve_june_5_2024.mw

Any idea why this sometimes happens? odeadvisor says ode is quadrature but when asking it to solve it using quadrature sometimes it works and sometimes not.

Am I doing something wrong?

Download sometimes_dsolve_works_on_quadrature_june_5_2024.mw

Any one could find why Maple 2024 gives Error, (in evalf/int) invalid arguments on this ode? Why is it even calling evalf in first place as this is all symbolic.

Also reported to Maplesof just in case.

Download odetest_internal_error_evalf_int_june_3_2024.mw

I wanted to do 

map(X->odetest(X,ode) assuming x>0,[sol]) 

where sol are the solutions I wanted to test. The above does not work as is. After some trial and errors, I found this works

map(X->[odetest(X,ode) assuming x>0],[sol]) 

I am not sure why. But it works. Is this the above the correct way to add assuming to a command inside Map? Or is there a different way or better way to do this?

I want to stick to the map command as above and nothing else such as ~ or some other ones. I find map more clear to use for me.

Here is worksheet.

Download how_to_use_map_with_assuming.mw