I'll attach two questions below Will rate if helpful! thanks

(1 point) We consider the non-homogeneous problem y\" y' = 60 cos(3x) First we consider the homogeneous problem y\" y = 0 : 1) the auxiliary equation is (1r2 + br + c = = 0. 2) The roots of the auxiliary equation are (enter answers as a comma separated list). 3) A fundamental set of solutions is (enter answers as a comma separated list). Using these we obtain the the complementary solution ya = c1y1 + czyz for arbitrary constants c1 and c2. Next we seek a particular solution y, of the non-homogeneous problem y\" y' = 60 cos(3x) using the method of undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find yp = 6cos(3x)+25in(3x) We then find the general solution as a sum of the complementary solution y; = c1y1 + c2y2 and a particular solution: y = yo + yp. Finally you are asked to use the general solution to solve an IVP. 5) Given the initial conditions y(0) = 8 and (0) = 7 find the unique solution to the IVP y = 2+600$(3x)+25in(3x) Help Sheet: Undetermined Coefficients Notes We consider the non-homogeneous problem y\" y' = 3 (60 cos(3x) + 4x) First we consider the homogeneous problem y\" y' = 0 : 1) the auxiliary equation is ar2 + br + c = = 0. 2) The roots of the auxiliary equation are (enter answers as a comma separated list). 3) A fundamental set of solutions is (enter answers as a comma separated list). Using these we obtain the the complementary solution ye = cly1 + czyz for arbitrary constants c1 and c2. Next we seek a particular solution yp of the non-homogeneous problem y\" y = 3 (60 cos(3x) + 4x) using the method of undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find yp = We then find the general solution as a sum of the complementary solution y; = c1y1 + czyz and a particular solution: y = yc + yp. Finally you are asked to use the general solution to solve an NP. 5) Given the initial conditions y(0) = 3 and y' (0) = 6 find the unique solution to the WP y: Help Sheet: Undetermined Coefficients Notes