Find the recurrence relation and the general term for the solution:
y'' - xy' - y = 0 xo=1
y= sum (n=0 to infinity) an (x-1)^n
The Attempt at a Solution
y= sum (n=0 to infinity) an x^n
y'= sum (n=0 to infinity) (n+1)an+1 (x-1)^n
y'' = sum (n=0 to infinity) (n+2)(n+1)an+2(x-1)^n
here all of the sums start at zero and the powers of (x-1) all are n, but within the orginal equation there is an +and this is where the textbook doesn't make sense:
the book sets x= 1+ (x-1). Why do you do this? i don't understand.
I thought that: xy' = sum (n=0 to infinity) (n+1) an+1 (x-1)^n+1
but i'm not sure what to do from here...bc the indexes and powers need to be the same before calculating the recurrence relation right?