1. The problem statement, all variables and given/known data (x^2)(y'') + xy' - 4y = 0 a) show that x^2 and x^-2 are linear independent solutions of this equation on the interval 0 <x<infinity b) write the general solution of the given equation c) find the solution that satisfies the conditions y(2) = 3, y'(2) = -1. Explain why htis solution is unique. Over what interval is this solution defined? 2. Relevant equations 3. The attempt at a solution ok im pretty sure i got part a). i just found y' and y'' for y = x^2 and plugged it in. then did the same for y = x^-2. both give me an answer of 0. it doesnt seem to matter what x is so i guess 0<x<infinity doesnt contain any false statement so i said it was true. for part b), im having trouble because either my professor didnt explain well or i took bad notes. the general equation is supposed to be: y(x) = Yc(x) + yp(x) i beleive Yc(x) = c_1(x^2) + c_2(x^-2) in the example in class yp(x) = 1/2 but there's no explanation on how we got that. and nothing in the book on "general equation" so any help on this would be appreciated for part c). i replaced x with 2, y with 3, and y' with -1 then i got y'' = 7/2. was that what i was supposed to do? then i was going to integrate back to find the equation of y that satisfies all the conditions. there's no clear explanation on this step either, so do i have the right idea for this?