With explanation pls, This is in preparation for my finals. Thank you!!

I. Multiple Choice: Choose the letter corresponding to the best answer. Write your answer and pertinent solutions. 1] Given f(D)y = x e2x sin3x + x2 . The roots of the auxiliary equation of yp, assuming yp is the particular solution of some homogeneous linear differential equation with constant coefficients are : (a) 2 + 31, 0 (b) 2 + 3i, 2 + 3i, 0,0 (c ) 2 1 3i, 2 + 3i, 0,0,0 (d) 2 + 3i, 2 + 3i, 2 + 3i , 0,0 2] The linear differential equation with constant coefficients satisfied by the equation y = 2x - 5ex cos5x is (a) (D3 + 4D2 + 29D ) y = 0 (b) (D4 - 4D3 + 29D2)y = 0 (c ) (D4 + 4D3 + 29D2)y = 0 (d) (D3 - 402 + 290 )y = 0 3] The auxiliary equation of the Linear Differential equation with real constant coefficients satisfied by the equation y =4sinh 3x is (a) m3 - m2 + 36m - 36 = 0 (b) m3 + 36m = 0 (c ) m3 - 81m = 0 (d) m3 - 36m = 0 4] The number of arbitrary constants in the general solution of the differential equation D3( D4 + 3D3 - 202 + 5)y = 0 is (a ) 3 (b ) 7 (c ) 4 (d ) 12 5] The particular solution of ( D4 ( D3 - 2D2 + 3D - 6)y = -16 is (a) yP= 3 (b) VP= 3x -x3 (c ) Up= = x4 (d) yp= = 6] In the differential equation (D2 + 4)y = cot 2x, using Variation of Parameters method, the proposed particular solution is (a) yp= A(x) e2x + B(x)e-2x (b) Up= A(x)cosx + B(x)sinx (c ) yp= A(x) ex + B(x)e-x (d) yp= A(x)cos2x + B(x)sin2x 7] The Wronskian of the functions 1, x, and e-2x is (a) e-2x(2x +1) (b) 4e-2x (c ) -e-2x (2x +1) (d) - e-2x(x+2)