Difference Equations, Part 2

Difference Equations

Part 2: First-Order Linear Difference Equations

Given a number a, different from 0, and a sequence {z_k}, the equation

y_k+1 + a y_k = z_k

is a first-order linear difference equation. If {z_k} 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 y_k from the preceding y-values. More specifically, if y₀ is specified, then there is a unique sequence {y_k} that satisfies the equation, for we can calculate, for k = 0, 1, 2, and so on,

y₁ = z₀ - a y₀,

y₂ = z₁ - a y₁,

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

y_k+1 = y_k + 0.00575 y_k - 450.

Explain why this equation is (or can be rewritten as) a first-order linear difference equation. What number is a? What sequence is {z_k}? Is the equation homogeneous or nonhomogeneous?

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 x_p of the nonhomogeneous system. The general solution of the nonhomogeneous system is then x₀+x_p. We will now carry out this program for the loan repayment problem.

Show that the sequence {y_k} defined by

y_k = (- a)^k

is a solution of the homogeneous equation

y_k+1 + a y_k = 0.

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.]

With a = -1.00575, find a constant solution of the nonhomogeneous equation

y_k+1 + a y_k = - 450,

that is, a solution for which every y_k is the same.

Find the general solution of

y_k+1 - 1.00575 y_k = - 450.

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?

Use your solution formula to find the month k in which the balance y_k goes from positive to negative. This should confirm the result you found in Part 1 for the time until the loan is paid off.

modules at math.duke.edu