Go Premium for a chance to win a PS4. Enter to Win

x
Solved

# Nonhomogeneous second order difference equation

Posted on 2007-11-20
Medium Priority
415 Views
x(n+1) - 4x(n) + 3x(n-1) = 36n^2

What form will the particular solution take?

I tried x(n)=an^2+bn+c, but when I plugged it in, I got a 0 coefficient for the n^2 terms on the LHS.
What is an alternative to try?

Thanks
0
Question by:Beta07
• 7
• 2

LVL 27

Expert Comment

ID: 20322017
It has been a while since I have worked with difference equations but the following two reference might be of help.

Recurrence relation - Wikipedia, the free encyclopedia
A difference equation is a specific type of recurrence relation. ... Certain difference equations can be solved using z-transforms. ...
en.wikipedia.org/wiki/Recurrence_relation - 50k - Cached - Similar pages

Difference Equations
Purpose: To apply linear algebra concepts to study the properties of sequences defined by difference equations. Prerequisites: The concepts of linear ...
www.math.duke.edu/education/ccp/materials/linalg/diffeqs/index.html - 5k - Cached - Similar pages

0

LVL 85

Accepted Solution

ozo earned 2000 total points
ID: 20322711
a*b^n + c*n^3 + d*n^2 + e*n + f
0

LVL 2

Author Comment

ID: 20322977
Interesting. The next question is:

x(n+1) - 4x(n) + 3x(n-1) = 3^n

Which I'm having the same problem with (getting 0 coefficients).

Would the particular solution take the form:

a*3^n + c*n^3 + d*n^2 + e*n + f

?
0

LVL 2

Author Comment

ID: 20323018
> a*b^n + c*n^3 + d*n^2 + e*n + f

Isn't that the form of the general solution?
And is there a reason for having a cubic rather than a quadratic? ("Because it works" will suffice)
0

LVL 85

Expert Comment

ID: 20323068
x(n+1) - x(n)  = 36n^2 would be cubic
x(n+1) - 4x(n) = 0 would be exponential
0

LVL 2

Author Comment

ID: 20323102
Ahhh, very clever!

I'll give it a shot, thanks
0

LVL 2

Author Comment

ID: 20323235
Hmm, when I plug

x(n) = a*b^n + c*n^3 + d*n^2 + e*n + f

into

x(n+1) - 4x(n) + 3x(n-1) = 36n^2

And equate coefficients, the only information I get out of it is:

d = 3c
c+d+e = 0

:-\
0

LVL 2

Author Comment

ID: 20324326
I think I may have figured it (this method has worked for 3^n instead of 36nÃ‚Â²), I'm just stuck on this one step;

The Shift Operator, E, is defined as:  E x(n) = x(n+1)
So obviously, E^k x(n) = x(n+k)

I need to define an annihilator A(E), such that:

A(E) n^2 = 0

An example for n is:

(E-1)^2 n = 0
0

LVL 2

Author Comment

ID: 20324623
Oh, lol

(E-1)^4  n^2 = 0

:-)

Which makes sense from more than one perspective ..
0

LVL 2

Author Comment

ID: 20324732
Ah, done it! :)

I used a slightly varied method to my previous attempts (obviously), but it practically paralleled ozo's solution; and in doing so, I realised where I went wrong with my first attempt with ozo's suggestion..

Thanks
0

## Featured Post

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Have you ever thought of installing a power system that generates solar electricity to power your house? Some may say yes, while others may tell me no. But have you noticed that people around you are now considering installing such systems in their â€¦
This article seeks to propel the full implementation of geothermal power plants in Mexico as a renewable energy source.
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.
Although Jacob Bernoulli (1654-1705) has been credited as the creator of "Binomial Distribution Table", Gottfried Leibniz (1646-1716) did his dissertation on the subject in 1666; Leibniz you may recall is the co-inventor of "Calculus" and beat Isaacâ€¦
###### Suggested Courses
Course of the Month9 days, 14 hours left to enroll