1. Use
showstat(DEtools:-odeadvisor);
or better:
showstat(DEtools[odeadvisor]);
because DEtools is a table-based package.

2.

stopat(DEtools[odeadvisor]);
 #                    [DEtools[odeadvisor]]
ode:=y(x)*(2*x^2*y(x)^3+3)+x*(x^2*y(x)^3-1)*diff(y(x),x)=0;
DEtools[odeadvisor](ode);

If possible, don't use floats in symbolic computation (here limit)

Compare:

limit(n*(1/3 - 1/(3+1/n)),n=infinity);
#                               1/9
limit(n*(1/3. - 1/(3+1/n)),n=infinity);
#                        -Float(infinity)

So, replace in W:  3.6  with 36/10  etc.

Of course, just use the menu File > Page Setup...
or Ctrl+Shift+P. 
Then Export or Print to a pdf port.

Probabilistic Algorithm Control - Maple Help (maplesoft.com)

restart;
FD:=proc(f::procedure, i::nonnegint, h::algebraic)# Finite Difference
    if i=0 then return f fi;
    x -> ( FD(f,i-1,h)(x+h) - FD(f,i-1,h)(x) )
    end proc:

f:= x -> x^5:
FD(f,3,h)(x): simplify(%);
seq( simplify(FD(f,i,1)(x)), i=0..6 );

          150*h^5+180*h^4*x+60*h^3*x^2

          x^5, 5*x^4+10*x^3+10*x^2+5*x+1, 20*x^3+60*x^2+70*x+30, 60*x^2+180*x+150, 120*x+240, 120, 0

odetest does not validate correct solution for first order ode - MaplePrimes

(Only two months ago)

Download solveZ.mw

These examples are useless. See also an algebraic equivalent:

restart;
eq:=sin(x)+x^4+x^2+x+1;
s:=solve(eq, x);      # RootOf(eq, x)
is(eval(eq, x=s)=0);  # FAIL

This seems to be related to the bug in the evaluation of 0^n.
See strange odetest result. Is this valid? - MaplePrimes

restart;
diff(1,x)^n;  # 0
0^n;          # 0^n
diff(1,x)^n;  # 0^n

Also for  diff(1,x) replaced with sin(0).

It's a bug about 0^x, not about odetest.

restart;
simplify(0^x) assuming x>0;
                 0^x
normal(0^x) assuming x>0;
                 0^x
eval(0^x) assuming x>0;
                 0^x
expand(0^x) assuming x>0;
                 0^x
0.0^x assuming x>0;
                 0.^x
#########################
is(0^x = 0) assuming x>0;
                 true
limit(0^x, x=x) assuming x>0;
                 0
eval(z^x, z=0) assuming x>0;
                 0
simplify(sin(0)^x) assuming x>0;
                 0
is(0.0^x = 0) assuming x>0;
                 true
#########################
eval(z^x, z=0) assuming x<0;   # ?
                 0

The bug seems to be rather old.
Also strange, in Maple 2018,  0^x  automatically simplifies to 0 (without any assumption).
I suspect that 0^x = 0 was removed from automatic simplification but the evaluation/simplification was not modified.
 

restart
dsolve( diff(y(x),x)=0, ['quadrature'])

Error, (in dsolve) numeric exception: division by zero

You asked a similar question recently.
In your examples dsolve simply does not eliminate a solution even when obtainable from another.
Anyway, note that actually:

•  (x + y(x))*(1+diff(y(x),x)) = 0 has one solution: y(x) = -x + c.
•  (x + y(x))*(a+diff(y(x),x)) = 0 has two solutions  (shown).
• (x + y(x))^2 *(1+diff(y(x), x))=0 has also one solution: y(x) = -x + c. 

If you express the ode in a standard form (as I told you in a comment a few days ago for a similar question),  ODESteps works.

restart;
ode1:=diff(y(x), x) = abs(x - 2)^(2/3);
ic:=y(2)=1;
Student:-ODEs:-ODESteps([ode1,ic]);

Note that the final result is actually 
y(x) = 1+3/5*abs(x-2)^(5/3)*signum(x-2);

Edit.
Maybe better (valid in complex too):
 

ode1:=diff(y(x), x) = ((x - 2)^2)^(1/3);

then:  

Your mysol is not in the form expected by odetest for an implicit solution: F(y(x), x) = 0; in this form everything is OK.

Of course, odetest could have converted mysol, but it does not; maybe it should.