Power series as a particular solution to a diff eq

[adanedhel728]

I've got this question that I've kind of hit a dead end on. The book that I have doesn't address it, because it's something that my professor handed out separately from the book, and my notes don't seem to cover it, or at least don't finish covering it. I've typed the whole thing up, including the work I've done so far, in MS Word's equation editor so I could post it on here.

[attachment=0:1ick7pa6]direct substitution.jpg[/attachment:1ick7pa6]

Yeah, it's really tedious. Unfortunately, I really don't know exactly what I'm doing, so I don't think I can really give more information than this. I think I'm trying to prove that (x+1)y''+y' is zero? If anyone can make sense of my work or can shed some light on the issue, I would really appreciate it. I've always been bad at series for some reason.

Thanks,
Andrew

 

[arthur ohlsten]

I shall do it by a brute force method. i suggest you do it by working with the series notation

y=x-x^2/2 + x^3/3 - x^4/4 ......

eq.1) y ' = 1-x+x^2-x^3+........

y ' ' = -1+2x-3x^2 + 4x^3
eq 2) [x+1] y ' ' = [-1+2x-3x^2+4x^3....]+[-x+2x^2-3x^3+4x^4]

then"
[x+1]y ' ' +y ' = [x-2x^2+3x^3-...] + [ -x+2x^2-3x^3+4x^4...]
[x+1]y ' ' + y ' = 0 proof

Arthur