|
|
|
Given a number a, different from 0, and a sequence {zk}, the equation
is a first-order linear difference equation. If {zk} is the zero sequence {0, 0, ... }, then the equation is homogeneous. Otherwise, it is nonhomogeneous.
A linear difference equation is also called a linear recurrence relation, because it can be used to compute recursively each yk from the preceding y-values. More specifically, if y0 is specified, then there is a unique sequence {yk} that satisfies the equation, for we can calculate, for k = 0, 1, 2, and so on,
and so on.
Example 1: Loan Repayment (continued). In Part 1 you found that the monthly loan balances could be calculated from the recurrence relation
Recall that we can separate the solution process for a linear system into two steps: First find the general solution x0 of the homogeneous equation. Next find one particular solution xp of the nonhomogeneous system. The general solution of the nonhomogeneous system is then x0+xp. We will now carry out this program for the loan repayment problem.
is a solution of the homogeneous equation
Explain why the general solution consists of all scalar multiples of this one solution. [Hint: Refer to the discussion above about an initial value determining a unique solution. Show that the solution space is one-dimensional.]
that is, a solution for which every yk is the same.
Then find the solution that satisfies the starting condition that y0 = 20000. What is it in this formula that tells you the balance must eventually be negative?
|
|
|
modules at math.duke.edu | Copyright CCP and the author(s), 1999 |