im ok with the given solution up to (4). i dont understand the method used to solve the particular solution. please can u name the method used and explain how to apply it? thank you

stion 3 Solve the system: (D' + 1)(2)] - 2D[:2) = 21, (20 - 1)[:] - (2 - D)(x2] = 5, by using the elimination method (operator method). Solution Given the system: (1) (D' + 1)(z]] - 2D[x2] = 2t (2) (20 - 1)(x]] - (2- D)[x2) = 5 Eliminating [x2] first, we calculate (2 - D) x (1) - (2D)[(2)]~ ~ to get: (3) (-D' - 203 + D + 2)(21] = 4t-2;0 From (3) we solve for I). The auxiliary equation is m3 + 2m? - m - 2 = (m - 1)(m 0 > m = #1, -2, yielding the complimentary function: (4) TIC.P. (1 ) = cle " + cze + ese" , v vv where ci, c2 and ca are arbitrary constants. The particular integral is given by: ( Note: You can also use the method of undeterm to find particular solutions) 4t - 2 TIPI.(1) = (D - 1)(D + 1)(D +2) (D + 2)(D - 1) - D + D2 - D3 +... (4t - 2) 1 (D +2) (D- 1) - (41 - 2 - 1) = (D + 2)(D _ 1) (4t -6) 1 = D +2 1 + D + D' + ... ( 4t - 6 ) D + 5(41 -6+4) = D+2 (4t - 2) -6 (4t - 2) 2 1 - 2+ 4 - ( 42 - 2) - 3(41 -2-2) - 141 i.e., (5) TIPS(1) = 21 -2.7 7