Seemingly Simple Difference Equation

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searcher8881

Joined Jan 17, 2017
1
For my linear systems class, I need to solve the difference equation for a fixed-rate mortgage using the Z-transform. I feel like a total idiot because I'm getting stumped by this. Any input would be appreciated.

Given:
P: principal
a: 1 + (interest rate/12)
b: monthly payment

The difference equation for the amount left to pay after k months is:

x(k+1) = ax(k) - b

Using the initial condition x(0) = P and the Z-transform method, solve for x(k) for k ≥ 0. Express your answer in terms of P, a, and b.


I did the Z-transform of this, yielding

X(z) - P = aX(z) - bz/(z-1)

Isolating X(z), doing partial fraction expansion on the second term, and then the inverse Z-transform, I get a function that doesn't really make sense.

x(k) = (P + ab/(1-a))*u(k-a) - (b/(1-a))*u(k-1)

Any idea what I'm doing wrong? Thanks.
 

wayneh

Joined Sep 9, 2010
16,400
I'm a little lost by the formatting. That and having no idea what a Z-transform is. In your first formula, everything in parentheses should be subscripted. But in the second formula there is the (z-1) term that I think is not subscripted. It gets worse from there.

This is an extremely over-complicated way to solve for the mortgage values. If the simple mortgage example is used to illustrate the z-transform concept, I guess that's fine. If your goal is to analyze a mortgage, this isn't the way to do it.
 
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