Answer Happy

Consider the variable coefficient linear non-homogeneous ODE a(x)y"' +b(x)y' + c(x)y = d(x) where ( 22 +2) cos(x) ) ) a(x) = sin(x)x – cos(x) x2 b(a) 7 x3 and - cos(x) x — 2 sin(x) C(20) = sin(x)x – cos(x) d(x) = x2 cos(x) 7 23 The two linearly independent solutions of the associated homogeneous equation are = Y1 cos(x), y2 = x-1 A particular solution to the non-homogeneous equation can be found using the method of variation of parameters, = Yp = uyi + vy2) where u and v are unknown functions. The solution method involves solving two first order ODEs for u and v.

(a) Which of the following is the expression for u!? cos(2)(sin(x)x-cos(x)) 22 sin(2)x-cos :(2) x2 sin(x )x-cos(2) X cos(x)(- sin(x)x+cos(x)) (b) Which of the following is the expression for v'? cos(2)(sin(x)x-cos(x)) x2 sin(x)x-cos(2) cos(2)(- sin(x)x+cos(x)) x2 sin(x)x-cos(x)