Difference Equation - Non Homogeneous need help

Hi

Can someone help me with this and provide a step by step response?

Suppose I have the following difference equation:

with

I have solved the characteristic eqn to be

But how do I go about solving the particular solution?

Many thanks in advance!

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**zzizi** Hi

Can someone help me with this and provide a step by step response?

Suppose I have the following difference equation:

with

I have solved the characteristic eqn to be

But how do I go about solving the particular solution?

Many thanks in advance!

Hi zzizi! :)

Wiki explains it better than I can:

Recurrence relation - Wikipedia, the free encyclopedia

The equation in the above example was [[homogeneous differential equation|homogeneous]], in that there was no constant term. If one starts with the non-homogeneous recurrence

with constant term ''K'', this can be converted into homogeneous form as follows: The [[steady state]] is found by setting

to obtain

Then the non-homogeneous recurrence can be rewritten in homogeneous form as

which can be solved as above.

Re: Difference Equation - Non Homogeneous need help

I would use the technique of **symbolic differencing** to obtain a homogeneous linear recurrence:

Subtracting the former from the latter, we obtain:

The characteristic roots are hence the closed form is:

We may use initial conditions to determine the parameters :

Solving this system, we find:

and so we have:

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**ILikeSerena** Hi zzizi! :)

Wiki explains it better than I can:

Recurrence relation - Wikipedia, the free encyclopedia

The equation in the above example was [[homogeneous differential equation|homogeneous]], in that there was no constant term. If one starts with the non-homogeneous recurrence

Thank you very much, for your reply.

So I was incorrect to think it was non-homogeneous?

I wasn't aware that it could be solved this way, so I will have to perhaps look into it further.

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**MarkFL2** I would use the technique of

**symbolic differencing** to obtain a homogeneous linear recurrence:

Subtracting the former from the latter, we obtain:

The characteristic roots are

hence the closed form is:

We may use initial conditions to determine the parameters

:

Solving this system, we find:

and so we have:

Thank you very much for this solution.

How would I go about checking it? I tried to find u(3) from the original recursion and the closed form but they didn't correlate.

Re: Difference Equation - Non Homogeneous need help

From the inhomogeneous recurrence you gave:

Using the closed form I gave:

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**zzizi** Thank you very much, for your reply.

So I was incorrect to think it was non-homogeneous?

I wasn't aware that it could be solved this way, so I will have to perhaps look into it further.

You were right. It is non-homogeneous.

In your case you have

Quote:

Originally Posted by

**zzizi** with

I have solved the characteristic eqn to be

If you compare that to

you'll see that you have:

As a result, your solution takes the form:

If you fill in your boundary conditions, you get 2 equations with 2 unknowns (C and D), which can be solved with substitution.

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**MarkFL2** From the inhomogeneous recurrence you gave:

Using the closed form I gave:

Umm ... I guess I miscounted (Giggle)

Thanks MarkFL2 :D I just saw your response on yahoo answers!!

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**ILikeSerena** You were right. It is non-homogeneous.

In your case you have

If you compare that to

you'll see that you have:

As a result, your solution takes the form:

If you fill in your boundary conditions, you get 2 equations with 2 unknowns (C and D), which can be solved with substitution.

Thank you very much for your explanation! Much appreciated.

Re: Difference Equation - Non Homogeneous need help

I have a query;

Is there another way, by substituting values to find the particular solution?

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**ILikeSerena** You were right. It is non-homogeneous.

In your case you have

If you compare that to

you'll see that you have:

As a result, your solution takes the form:

If you fill in your boundary conditions, you get 2 equations with 2 unknowns (C and D), which can be solved with substitution.

What if I end up with zero for the b* value? I used this method to solve my difference equation but I got zero for the constant. If this happens would the whole solution be complete or do I try another method?

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**zzizi** What if I end up with zero for the b* value? I used this method to solve my difference equation but I got zero for the constant. If this happens would the whole solution be complete or do I try another method?

The only way that b* can be zero is if the constant K is zero.

In that case the difference equation is a homogeneous difference equation instead of an in-homogeneous one.

But this is not applicable to your current problem statement.....

Am I misunderstanding you?

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**ILikeSerena** The only way that b* can be zero is if the constant K is zero.

In that case the difference equation is a homogeneous difference equation instead of an in-homogeneous one.

But this is not applicable to your current problem statement.....

Am I misunderstanding you?

You are quite right. I wanted to understand the concept because I have another problem for my homework that I am working which is like this:

When I applied the formula I got this:

SO would this be considered a homogeneous D.Eqn?

Re: Difference Equation - Non Homogeneous need help

Quote:

Originally Posted by

**zzizi** You are quite right. I wanted to understand the concept because I have another problem for my homework that I am working which is like this:

When I applied the formula I got this:

I just wondered what I should do in this situation

I'm afraid you miscalculated b*.

Either way, it means this method does not work.

Next method in line is the *symbolic differentiation* method MarkFL2 described.

That one works for this problem.

Re: Difference Equation - Non Homogeneous need help