Consider the variable coefficient linear homogeneous ODE a(x)y" + b(x)y' + c(x)y = 0 where sin(z) (z² +2) sin(x)x – 2 cos(x) c(x) = 22 (cos(x)x + sin(æ)) a(x) = x-1, b(x) 2² (cos(x)x+sin(x)) A solution of the equation is y1 = sin(x). A second linearly independent solution can be found using reduction of order Y2 = UY1, where u(x) is an unknown function. The solution method involves solving a first order ODE for w(x) which determines the unknown function u() by solving another first order ODE w(x) = u'(x). Which of the following is the expression for w? sin(=) (-2 +2) - 2 cos(x) sin(x) w z(cos(x)x+sin(x)) sin(z) (z² +2) cos(x) +2 sin(x) w ¤(cos(x)x+sin(x)) sin(2)(z² +2) +2 cos(x) sin(x) w æ(cos(x)x+sin(x)) sin(x) a(=) (2²+2) cos(x) sin(x) w æ(cos(x)x+sin(x))

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