January 28th, 2012, 10:24 PM  #1 
Newbie Joined: Jan 2012 Posts: 5 Thanks: 0  changing difference equation
I have to change this systems of difference equation to a single higher order equation. The answer is What I did so far was set it equal to zero and got this: I am not sure what to do after. I think I am supposed to find the characteristic equation but my equation doesn't look like a polynomial. Can someone help me figure this out please? Thank you for your time! 
January 28th, 2012, 10:27 PM  #2 
Newbie Joined: Jan 2012 Posts: 5 Thanks: 0  Re: changing difference equation
Whoops. I made a mistake on the second equation. It is 
January 28th, 2012, 11:01 PM  #3 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,204 Thanks: 511 Math Focus: Calculus/ODEs  Re: changing difference equation
We are given: (1) (2) Method 1: Taking (2) and replacing n with n + 1 gives: Using (1), substitute for : Solving (2) for and substituting, we then get: Now we have a linear homogeneous recurrence, whose associated characteristic equation is: Thus, we find the closedform for is: where A and B are arbitrary constants. Method 2: Taking (1) and replacing n with n + 1 gives: Using (2), substitute for : Solving (1) for and substituting, we then get: Now we have a linear homogeneous recurrence, whose associated characteristic equation is: Thus, we find the closedform for is: where A and B are arbitrary constants. 

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