Solved

Nonhomogeneous second order difference equation

Posted on 2007-11-20
10
402 Views
Last Modified: 2008-02-01
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
Comment
Question by:Beta07
  • 7
  • 2
10 Comments
 
LVL 27

Expert Comment

by:aburr
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 84

Accepted Solution

by:
ozo earned 500 total points
ID: 20322711
a*b^n + c*n^3 + d*n^2 + e*n + f
0
 
LVL 2

Author Comment

by:Beta07
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

by:Beta07
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 84

Expert Comment

by:ozo
ID: 20323068
x(n+1) - x(n)  = 36n^2 would be cubic
x(n+1) - 4x(n) = 0 would be exponential
0
Enabling OSINT in Activity Based Intelligence

Activity based intelligence (ABI) requires access to all available sources of data. Recorded Future allows analysts to observe structured data on the open, deep, and dark web.

 
LVL 2

Author Comment

by:Beta07
ID: 20323102
Ahhh, very clever!

I'll give it a shot, thanks
0
 
LVL 2

Author Comment

by:Beta07
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

by:Beta07
ID: 20324326
I think I may have figured it (this method has worked for 3^n instead of 36n²), 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

by:Beta07
ID: 20324623
Oh, lol

(E-1)^4  n^2 = 0

:-)

Which makes sense from more than one perspective ..
0
 
LVL 2

Author Comment

by:Beta07
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

Find Ransomware Secrets With All-Source Analysis

Ransomware has become a major concern for organizations; its prevalence has grown due to past successes achieved by threat actors. While each ransomware variant is different, we’ve seen some common tactics and trends used among the authors of the malware.

Join & Write a Comment

Suggested Solutions

Title # Comments Views Activity
Caveats, pros & cons of going on CNG for automobiles 2 77
2k Power n formula 2 33
Inequality 4 58
Dual bridge protection 18 46
A Guide to the PMT, FV, IPMT and PPMT Functions In MS Excel we have the PMT, FV, IPMT and PPMT functions, which do a fantastic job for interest rate calculations.  But what if you don't have Excel ? This article is for programmers looking to re…
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 …
It is a freely distributed piece of software for such tasks as photo retouching, image composition and image authoring. It works on many operating systems, in many languages.
This video shows how to remove a single email address from the Outlook 2010 Auto Suggestion memory. NOTE: For Outlook 2016 and 2013 perform the exact same steps. Open a new email: Click the New email button in Outlook. Start typing the address: …

708 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question

Need Help in Real-Time?

Connect with top rated Experts

18 Experts available now in Live!

Get 1:1 Help Now