Difference Equations
Part 3: Second-Order Equations
Given numbers a1 and a2, with a2 different from 0, and a sequence {zk}, the equation
yk+2 + a1 yk+1 + a2 yk = zk
is a second order linear difference equation. If {zk} is the zero sequence {0, 0, ... }, then the equation is homogeneous. Otherwise, it is nonhomogeneous.
Example 2: A National Economy (continued). The model presented in Part 1 for a national economy involved the equation
yk+2 - a(1 + b) yk+1 + ab yk = 1.
- Explain why this model equation is a second order linear difference equation. What are the coefficients a1 and a2? Is the equation homogeneous or nonhomogeneous? Why?
We saw in Part 1 that solutions of this model have rather different behavior from the solutions of Example 1 -- in particular, in Example 2 we saw convergence to an "equilibrium state" that was apparently independent of the starting conditions. We explore now the sequences that can be solutions of second order difference equations.
- Certain sequences {yk} of the form
yk = rk
are solutions of the homogeneous equation
yk+2 + a1 yk+1 + a2 yk = 0.
What condition must be satisfied by r in order for {rk} to be a solution?
- If r1 and r2 are different numbers that satisfy the condition in the preceding step, explain why the sequences {r1k} and {r2k} are linearly independent.
- For a = 0.9 and b = 0.5, as in Part 1, calculate a1 and a2. Then calculate r1 and r2. Finally, calculate the general solution of the homogeneous equation
yk+2 - a(1 + b) yk+1 + ab yk = 0.
- With the given values of a and b, find a constant solution of the nonhomogeneous equation
yk+2 - a(1 + b) yk+1 + ab yk = 1.
Then find the general solution of the nonhomogeneous equation.
- Explain why every solution must converge to the constant solution you found in the preceding step.
- What feature of the solutions of the loan repayment model (Part 2) makes those solutions diverge? [Hint: The answer has nothing to do with the order.]
