use reduction of order to find general solution of x^2y''+xy'+(x^2-(1/4))y=0 , (y1(x)=(sin x)/(sqrt x))

Answer 1

Similar to your other reduction of order question. Let y2(x) = y1(x)v(x) for some v. Substitute this into the equation and get

x^(3/2)sin(x) v'' + 2x^(3/2)cos(x)v' = 0.

Let w = v', and divide through by x^(3/2)sin(x), and get

w' + 2cot(x)w = 0

meaning w = exp(-2ln(sin(x)))w0 = csc^2(x)C1, where C1 is a constant. Integrate this to get v = -C1*cot(x) + C2, where C2 is also a constant.

Answer 2

I am in full agreement with and fully endorse what Mr."Aryan" has stated in his answer which is not only very detailed but very precise ,to the point and very crisp too.