Input in Differential Equations and Difference Equations

I am a Mathematica user and know the history of the package rather well. They introduced the difference equations in version 7.0 and the differential equations in the first. So there is a big difference in how interested people are in either.

In DifferenceRoot there is last section that is of high interest to the intent of Your question:

GeneratingFunction[
 DifferenceRoot[
   Function[{y, n}, {-y[n] - y[n + 1] + y[n + 2] == 0, y[0] == 0, 
     y[1] == 1}]][n], n, z]

$$-\frac{z}{-1 + z + z^2}$$

The built-in is DifferenceRoot. The difference equation in this example is the second list to the function. This is in terms of Mathematica Your similarity between difference equation and differential equation. There is not much.

Differential equation are special cases of limit of a difference quotient over time solely. Difference equation are part of the theory of generating functions GeneratingFunction. There are differential equation for generating functions, GeneratingFunction, and difference equations. Both define the generating function complete.

Most generating functions are rational polynoms or transcendential functions.

There are too exponential generating functions:

Both are mathematics but very different in nature. Do not mix them up.

Mathematica knows some more of this functions categories infinite sums and recurrence equations.

The solution to difference equation are holonomic sequences. The solution to differential equation do not necessarily need to suit more than differentiability of suitable order.

But this is only the Mathematica version. Difference equation appears in very different contexts.

Examples:

1. Difference_equation as a scheme to set up solvers for differential equations.

2. The above detailed Mathematica interpretation deals with homogeneous differential equations.

3. Defnition for scheme for interpolation of functions or data.

4. The equations in school math with at least one subtraction.

5. This introduction the relation as in Mathematica with recurrence, What-is-a-difference-equation. ....

   sol = RSolve[{y[n + 2] - 7 y[n + 1] + n y[n] == 0, y[0] == 5, y1 == 10}, y, n]

give a solution in Mathematica. With f there is non.

This is very rapidly increasing.

This shows the homogeneous function diverges much faster than the inhomogenity does and is pausible that this might be an error in question.

I made use of What is a differnce equation? on google.com