Is finding the solution of;
$\displaystyle
(D^2+10D+25)y = 0
$
The same as solving;
$\displaystyle
y''+10y'+25y = 0
$
??
Agreed.
I could not quickly find an example of D being used for a difference operator, at least not in the texts available to me. This surprises me. Maybe I dreamed up that I had seen it a time or two.
In any case, it is important to point out that notation is not always consistent. There is no substitute for the definitions provided in the discussion where the symbol is introduced.