Consider the variable coefficient linear homogeneous ODE a(x)}" + b(x)Y' + c()y=0 where a(x) = x-1, b(x) sin(x) (22 +2)

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Consider The Variable Coefficient Linear Homogeneous Ode A X B X Y C Y 0 Where A X X 1 B X Sin X 22 2 1 (143.31 KiB) Viewed 28 times

Consider the variable coefficient linear homogeneous ODE a(x)}" + b(x)Y' + c()y=0 where a(x) = x-1, b(x) sin(x) (22 +2) 22 (cos(2)x+ sin(2)) sin(x) .- 2 cos(2) c(x) = 22 (cos(2)x+ sin(x)) 9 A solution of the equation is yı = 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(2) which determines the unknown function u(x) by solving another first order ODE w() = u(x). Which of the following is the expression for W' ? sin(x)(x2+2) -2 cos(2) sin(2) -). w x(cos(2)x+sin(2) sin(x) (x2+2) +2 (cos(x)x+sin(x)) cos(3) sin(2) w sin(x)(x2+2) + 2 cos(x) sin(x) (cos(x)x+sin(x)) w sin(x)(22+2) <(cos(2)x+sin(x)) cos(2) - 2 sin() .). w