Mathematical methods for economic theory

Martin J. Osborne
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9.2 Exercises on second-order difference equations

  1. Solve the following difference equations and determine whether the solution paths are convergent or divergent, oscillating or not.
    1. xt+2 + 3xt+1 − (7/4)xt = 9.
    2. xt+2 − 2xt+1 + 2xt = 1.
    3. xt+2 − xt+1 + (1/4)xt = 2.
    4. xt+2 + 2xt+1 + xt = 9·2t.
    5. xt+2 − 3xt+1 + 2xt = 3·5t + sin((1/2)πt).

    Solution

    1. A1(1/2)t + A2(−(7/2))t + 4. Nonconvergent oscillations.
    2. (√2)t(A1cos (π/4)t + A2sin (π/4)t) + 1. Nonconvergent oscillation.
    3. A1(1/2)t + A2t(1/2)t + 8. Convergent, non-oscillating.
    4. The characteristic equation is m2 + 2m + 1 = (m+1)2 = 0, which has a double root of −1. So the general solution of the homogeneous equation is xt = (C1 + C2t)(−1)t. A particular solution is obtained by inserting ut* = A2t, which yields A = 1. So the general solution of the inhomogeneous equation is xt = (C1 + C2t)(−1)t + 2t.
    5. By using the method of undetermined coefficients the constants A, B, and C in the particular solution u*t = A5t + Bcos (π/2)t + Csin (π/2)t, we obtain A = 1/4, B = 3/10, and C = 1/10. So the general solution to the equation is
      xt = C1 + C22t + (1/4)5t + (3/10)cos((π/2)t) + (1/10)sin((π/2)t).