PG,
Q1. The author wants to emphasize that the assumed and soon to be found function u(x) is a solution of the differential equation x²y''-6y=0. That means x≠0. This is important for the reduction of order method.
Q2. The calculated integrating factor is x^6. Multiplying the equation by x^6 will make the left side of the equation into a complete differential. Therefore the solution of the equation is w=c/x^6 . Try substituting it into w'+6w/x = 0 and see.
Q3. The author could have chosen any constant values for c1 and c2 to get y2. Try different values of c1 and c2, then substitute into x²y''-6y=0 and see. He chose those (-5,0) values to get the simplest and lowest degree term value for y2.
Ratch