# Two questions about differential and difference equations

• posted

I'm working on an instructional video, and while I don't mind tossing out some hand-waving, and even some minor inaccuracies, I don't want to get things ENTIRELY wrong.

First question:

Quite a while ago, quite by accident, I discovered that one could solve linear shift-invariant difference equations via "the hard way" in much the same way that one solved linear differential equations.

In other words, given

y_k + a_1 * y_{k-1} + a_0 * y_{k-2} = f(k)

then one could solve the thing by first finding the homogeneous solution (arrived at by setting f(k) = 0), and then finding the homogeneous solution (assuming that f(k) was, by chance or design, equal to something "easy", i.e. some sum of A * k^n * d^m).

All of this was done by direct analogy to how it's done with differential equations.

Does anyone have any references to this? It can't be entirely my own invention. Even a name that coughs up results in a Google search would help.

Second question:

When you come up with the solution to a non-homogeneous solution (f(k) !=

0, above), the the range of functions that f(k) can be to keep the problem "easy" is constrained to functions that have "easy" z transforms (i.e., some limited sum of A * k^n * d^m). Is there a theorem associated with that? Again, references or a name I can Google (really, a name that I can set my viewers to Googling) would be cool.

If I don't get any answers in a day or a few I'm just going to hand-wave, and leave people puzzled. But I'd much rather be able to point them in a fruitful direction.

```--
Tim Wescott
Wescott Design Services ```
• posted

I'm working on an instructional video, and while I don't mind tossing out some hand-waving, and even some minor inaccuracies, I don't want to get things ENTIRELY wrong.

First question:

Quite a while ago, quite by accident, I discovered that one could solve linear shift-invariant difference equations via "the hard way" in much the same way that one solved linear differential equations.

In other words, given

y_k + a_1 * y_{k-1} + a_0 * y_{k-2} = f(k)

then one could solve the thing by first finding the homogeneous solution (arrived at by setting f(k) = 0), and then finding the homogeneous solution (assuming that f(k) was, by chance or design, equal to something "easy", i.e. some sum of A * k^n * d^m).

All of this was done by direct analogy to how it's done with differential equations.

Does anyone have any references to this? It can't be entirely my own invention. Even a name that coughs up results in a Google search would help.

Second question:

When you come up with the solution to a non-homogeneous solution (f(k) !=

0, above), the the range of functions that f(k) can be to keep the problem "easy" is constrained to functions that have "easy" z transforms (i.e., some limited sum of A * k^n * d^m). Is there a theorem associated with that? Again, references or a name I can Google (really, a name that I can set my viewers to Googling) would be cool.

If I don't get any answers in a day or a few I'm just going to hand-wave, and leave people puzzled. But I'd much rather be able to point them in a fruitful direction. ================================================================

Way, way back (35 yrs, sigh) I used the book "Continuous and Discrete Signal and System Analysis" by McGillem and Cooper in my signals and systems course. In the chapter on z transforms there is a little section on solving difference equations with z transforms that definitely covers your first question. Sorry, but it's been long enough since I looked at this that I'm not going to think hard enough to make sure they cover your second question, but a casual glance looks promising :-). So, I'll bet you can find it in any standard sig and sys text.

----- Regards, Carl Ijames

• posted

Here are two:

1. "Digital Signal Processing" by Proakis and Manolakis
2. "Numerical Methods for Scientists and Engineers" by Richard Hamming

I would say all this falls out of a z-transform analysis. The system function H(z) = Y(z)/F(z) is the solution of the homogeneous problem. The solution of the non-homogeneous problem is Y(z) = H(z)F(z) in the z-domain and y() therefore is a convolution of h(k) and f(k). This is why the solution has a factor that resembles the right-hand side.

• posted

I'm trying to do this _without_ introducing the z transform. It's a 15- minute talk; not 15 hours.

```--
Tim Wescott
Wescott Design Services ```

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