@ruhamdam 
 

D(y):=1: D(x):=x*y: D(z):=z^2: D(a):=0:
D(x*y + y*z + a*w);

      a*D(w) + x*y^2 + x + y*z^2 + z

@nm So, do you think that mathematically

LIMIT( diff(y(x), x) / y(x), y(x)=0) = 1   # ?

I have mentioned what odetest did. It is wrong.

[Edit] A math example.
Take y[n](x) = exp(-n*x), x in [1,2].
Then y[n] --> 0 (uniformly wrt x), but y[n]' / y[n] --> - oo.

@mmcdara SumOfDivisors is much older than 2015: numtheory:-sigma

@nm It is not Q which may fail (as you see, it is very simple). The problem is inherent for any CAS which cannot recognize sometimes a constant (or even 0) expression. Anyway, Q is much more reliable than solve. It is possible to use verify to check whether the result is constant or just evaluate it for some constants x,y (e.g. x=1, y=1 and then x=1,y=2) to obtain p; if p is the same, then check it directly; if not, then f is not isobaric.

@nm It works after a proper simplification:

Q((-(x*y)/2+sqrt(x^2*y^2-4*y)/2)*y, x,y):
expand(rationalize(%));
#                               -2

Edit. Q fails only for f(x,y) of the form f(x,y) = g(x)*y, 
but then g  must be g(x) = A/x, so, only the functions f(x,y) = A*y/x are not identified.

@nm The values given by P are correct too. 

restart;
f:=(x,y)->sqrt(x*y):
'p'=P(f(x,y), x,y):
p:=-1;
t^r=simplify(f(t*x, t^p*y)/f(x,y)) assuming t>0;
p:=2;
t^r=simplify(f(t*x, t^p*y)/f(x,y)) assuming t>0;
#                            p := -1
#                            t^r  = 1
#                            p := 2
#                            t^r  = t^(3/2)     

Actually, in this example p can be arbitrary!

@AHSAN I think you don't understand what I mean. When you try to compute e.g. int(x+y, x), you may obtain any of the expressions  x^2/2 + x*y,  x^2/2 + x*y + 7, x^2/2 + x*y + y^2 - ln(y),  x^2/2 + x*y + y^12 + sinh(y^2-2)  etc.
How are you going to interpret the plot, without knowing which expression is plotted?

 

@AHSAN I have told you that int(v,x) cannot be expressed by elementary functions. Maple probably tries to find a (huge and useless for you) expression containing sums over RootOfs (based on the residue theorem).
But even if B existed, it would be only modulo a function of y, so, your plot does not make much sense.

@AHSAN It's not the same thing. The new v is a simple rational function in x which has an elementary antiderivative.

@AHSAN  Sorry, I do not understand. Be more specific.

@DanishMapleFan  For example, if g is even and h is not even then f does not exist.

To adapt for earlier versions it is (probably) enough to replace a += b with a := a + b,
To map into [0,1]^2  just divide the result of d2xy  by n.
For an arbitry dimension, more work is needed, and I do not have the time for this. Note that it is easy to implement a Lebesgue filling curve in any dimension (just a few lines in Maple).

@acer In fracdiff, alpha should be not an integer. For alpha a float integer (such as 0. and 1.), fracdiff returns 0. I wonder whether this is intentional.

@JAMET The answer contains a numerical example. 

@kiraMou Ok, I see now that you are using Maple 18. Please replace that line with

print(Sol[crt] = eval(X, SOL[i][j])); crt:=crt+1;

I hope it will work for you too.