Can you help me express x(k) ?

Question

Given

x(k+2) = ax(k+1)+bx(k)
+ce(k+1)+de(k)

for all integer k > 0, where a, b, c, and d are known constants.

Express x(k) in terms of x(1), x(2), e(1), e(2)..., e(k-1).

Answer 1

Here's a good start.

Let X be the infinite sequence of x_k's and E be the infinite sequence of e_k's. Define x*an infinite sequence to be a shift of everything one to the right. (x_k becomes x_k+1). Then the above equation reads:

(x^2-ax-b)*X = (cx+d)*E.

Now suppose that p(x)=x^2-ax-b is not a perfect square (b^2<>4ac). Then the two roots r and s of p(x) are distinct. Also let F be the infinite sequence (cx+d)*E. The equation becomes

(x-r)(x-s)*X = F.

We can find X if we can solve (twice) an equation of the form:

(x-m)*X = G for arbitrary m and G.

When you write out what this means, it has a simple iterative solution.

x_2 = g_1 + m x_1,

x_3 = g_3 + m g_2 + m^2 g_1 + m^3 x_1,

etc.

I realize this is probably not what you wanted, but it gives a nice structural way to look at it.