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Difference Equations

Part 6: Summary

  1. Describe in your own words a first-order linear difference equation. Repeat for second-order and third-order.

  2. What kind of sequences {yk} do we know can be solutions of homogeneous linear difference equations? (There are other types, but only one type turned up in this module.) How are these sequences related to the coefficients in the difference equation?

  3. For the sequences identified in the preceding question, what determines whether these sequences converge or diverge as k becomes large?

  4. Suppose the characteristic polynomial of a third-order homogeneous difference equation has three distinct roots, r1, r2, and r3. How do you find a basis for the solution space of the equation? Describe the general solution.

  5. Suppose {qk} is a known solution of a third-order nonhomogeneous equation whose corresponding homogeneous equation is described in the preceding question. Describe the general solution of the nonhomogeneous equation.

  6. Suppose the general solution of a linear difference equation has the form

    yk = c1 (0.5)k + c2 2k + k2.

    • What is the order of the equation?
    • Is the equation homogeneous or nonhomogeneous? Why?
    • What is the characteristic polynomial of the equation?
    • What is the unique solution that satisfies y0 = 1, y1 = 0?

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