My question is:

Is there a duality between a solution of an ODE,PDE,SDE or integral equations with their analog counterpart in the discrete domain?

I mean if I know a solution to the difference equation will that mean that I can find a solution to the differential equation and vice versa?

In the case of well behaved, analytic solutions I believe the answer is yes, cause I can insert a power series as the solution and look for a difference equation which suits the differential equation, but what happens when we can't guess for an analytic solution is the continuous domain, can we still find a discrete counterpart to our DE?

There's one book which I found at my university library that I lend but still didn't find the time to read it thoroughly, it's called "Differential-Difference Equations" by Richard Bellman and Kenneth Cooke, I'll start reading next week, hopefully I'll have time for it.