Question

(1 point) We consider the non-homogeneous problem y" - y' = -4 cos(x) First we consider the homogeneous problem y -y = 0 : = 0 1) the auxiliary equation is ar2 + br + c = 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 = ciyı + c2y2 for arbitrary constants c1 and c2 Next we seek a particular solution y, of the non-homogeneous problem y" - y = -4 cos(x) using the method of undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find y, = We then find the general solution as a sum of the complementary solution ye = ciyi + c2y2 and a particular solution: y = yc + yp. Finally you are asked to use the general solution to solve an IVP 5) Given the initial conditions y(0) = 2 and y (0) = 1 find the unique solution to the IVP y= Help Sheet: Undetermined Coefficients Notes

Answer 1

Given differential equation is y-y' = -4COSX D the auxillary equcobon is 2-Y = 0 2) the roots of quxillary equabon are 10,1 (zeso g one) 3) the fundamentul set of solukon is [ai, cze? Complemendor :lyc = c + Core / solution is." let yp - A cosa + Bring 2 yp = - Asinx + B Cosy I 9'' = -AcOSX - Bsinx J use this in differential equation Now Yo' - 4P = -4605 31

- A (off – Bìnx + + Bình - Bosx = -( ( JY = cosx f-A-B] e sinx -6+A] = -4 cost = -A-B=4 A-B =0 solving this we get [B= 2] , [A=2 19 2 ( x + 2sinx @ general solution is f(x) = Cif Czé + 2CO3y + zsin y'(x) = (zett zsinx + 2005X 9(0)=2 = C1+(2+2=2 = (1+(2=0 '(0) = 1 = (2+2 = 1 1227 :.[11] © Unique solution for the IUP is α -ě + 200SX +25ÌNX | Thank you.