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MATH 2341 QUIZ 4A FALL 2022 NAME _ SOLUTIONS. October 6, 2022 (25 POINTS) NO CREDIT GIVEN UNLESS THE WORK IS SHOWN. 1 Given the equation 2y" + 6y’ = fx) (A) Find ». (1.5 points) The characteristic equation is 27 + 67 = 0 + 2r (r+3)=0 > rr " 2s w Therefore, ye = cr + @e* OR Hz ce* + (B) Determine the form of yy if the Ax) in (A) has each of the following values. Do not solve for the values of the yp coefficients. (i) fe) = & +5) e* (2 points) dp = x(Ax + B)e* (Case 2: a = -3 is a root once) (i) fe) = x+ Se (3 points) Jp = x (Ax +B) + Cre* (Case 1: 0 isa root and Case 2: a = -3is a root once) (iii) fx) = xeé + § (3 points) ¥p = (4x + Bye’ + Cx (Case 2: a = Lis nota root and Case I: 0 is a root) 2 Given the equation y" + 125y = f(x) (A) Find ».. (1.5 points) The characteristic equation is 7 + 25=0 3 r= +5; Ve = ce cos Sx + ep sin 5x (B) Determine the form of yp if the Xx) in (A) has each of the following values. Do not solve for the values of the yp coefficients. (i) fix) = 3co0s 5x (2 points) Yp = x (A cos 4x + Bsin 4x) (Case 3: a=0,6=5 and ati =0+5/ are the roots, so k= 1) (ii) fx) = 3cos 6x- sin 6x (2 points) Jp = Acos6x + Bsin 6x (Case 3: a= 0,6=6and a + bi =0+6) are not the roots once, so k= 0)