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Nonhomogeneous difference equation question

  1. May 9, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the general solution

    2. Relevant equations
    x(t+2)-3x(t+1)+2x(t)=3*5^t+sin(0.5πt)

    3. The attempt at a solution
    I start out by solving the homogeneous equation and end up with the two roots 1 and 2.

    Then I try to use the method of undetermined coefficients to find a particular solution.

    I guess that the solution is of the following form (I should probably include a cos term as well)

    u(t)=C*3*5^t+D*sin(0.5πt)

    then

    u(t+1)=C*3*5^(t+1)+D*sin(0.5π(t+1))

    and

    u(t+2)=C*3*5^(t+2)+D*sin(0.5π(t+2))

    I then insert these equations into the original equation to get

    C*3*5^(t+2)+D*sin(0.5π(t+2))-3*(C*3*5^(t+1)+D*sin(0.5π(t+1)))+2*(C*3*5^t+D*sin(0.5πt))=3*5^t+sin(0.5πt)

    move the terms around a bit to get

    3*5^t(C*5^2-3*C*5+2*C)+D*sin(0.5π(t+2)-3*D*sin(0.5π(t+1))+2*D*sin(0.5πt)=3*5^5+sin(0.5πt)

    From the first part I see that

    (C*5^2-3*C*5+2*C)=1

    so

    C=1/12

    I would like to do something similar for the sin part of the expression, but I'm not sure how to handle it.
    I tried with sin(0.5π(t+2))=sin(0.5πt+π)=-sin(0.5πt) but then I don't know what to do with sin(0.5(t+1)).

    If anyone has any hints, tips or tricks I would be happy.

    I hope the equations are understandable otherwise I'll post them as a picture.

    Thanks in advance.
     
  2. jcsd
  3. May 9, 2012 #2

    sharks

    User Avatar
    Gold Member

    Is this your original equation? It doesn't look like a non-homogeneous ODE to me.
    [tex]x(t+2)-3x(t+1)+2x(t)=3(5^t)+sin(0.5πt)[/tex]
    I just noticed in your title you call it a "difference equation" and not "differential equation", in which case, i might have misunderstood the problem.
     
  4. May 9, 2012 #3
    I just realized I can do subscripts. The original problem is this:

    xt+2-3xt+1+2xt=3(5t)+sin(0.5πt)

    Yeah it's a difference equation so we only have discrete moments of time.
     
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