. Solve the difference equation for the expected time to absorption of a simple random walk at 0 , pD^ , , -D^ + qD,_ 1 = - 1, with boundary conditions Dq = D^ = 0. To do this proceed as with the solution of nonhomogeneous second-order differential equations with constant coefficients. Thus, write the general solution as

'(p)

where is the general solution of the homogeneous equation (see Exercise 7) and D^p, is a particular solution of the nonhomogeneous equation. The appropriate guess for is ka where /c is a constant found by substitution. The arbitrary constants in the general solution are obtained by applying the boundary conditions. Obtain the solution in the special case p = q = \ by making a limiting argument as in Exercise 6 .