Given the partial solution to a problem. I want you to please explain to me how to determine the particular solution in the given example. Which equation to use when determining the particular solution. Do not solve the problem, just explain the particular solution where the process of undetermined coefficients are used.

Example 1.16 Solve the system (D + 1) [x] - (D + 1) [y] = et (D - 1) [x] + (2D + 1) [y] = 5 2 by using the elimination method (operator method). Eliminate y first. Solution Eliminating y first we find: (2D + 1) [(1)] : (2D2 + 3D + 1) [x] - (2D2 + 3D + 1) [y] = (2D + 1) [et] = 3et (3) (D + 1) [(2)] : (D2 - 1) [x] + (2D2 + 3D + 1) [y] = (D + 1) [5] =5 (4) (3) + (4) : 3D (D + 1) [a] = 3et + 5. (5)From the auxiliary equation m (m + 1) = 0 we have m = 0 or m = -1 so that the complementary function is given by XC.F. (t) = c1+ cze-t with ci and c2 arbitrary constants. We find the particular integral xp.I. (t) by using the method of un termined coefficients: Let XP.I. (t) = Aet + Bt so that acp.I. (t) = Aet + B XCP.I. (t) = Aet.Substituting these equations in (5) we find 6Act + 3B = 3et + 5. Comparing coefficients we find A = NIH and B = so that XP.I. (t ) = e t