I'm running into some confusion regarding my ODE book when dealing with Power Series Solutions; in the very basic examples of expressing y, y', y'' as power series, the indices seem to be changing, and I don't understand why. Screencap below momentarily.

Question

mendicant_bias · Answer

http://i.imgur.com/4hHnDVO.png Why is the lower index on these summations changing from zero to one to two? My line of thinking would be that the derivatives eliminate the first or second terms and thus they just raise the index entirely to avoid those terms being needlessly calculated, but in the paragraph before the blue formulas, they still have the lower indices for y' and y'' as zero.

ganeshie8 · Answer

the terms are 0 when n=0 or 1, right ?

ganeshie8 · Answer

for y''

mendicant_bias · Answer

One sec, figuring out what you mean. Oh, yeah, yup, that's what I was talking about when I meant I get why they could raise the indices to account for that, but was confused about the inconsistency between the two.

It might just be chronologically ordering things and that that's the way it's supposed to be, I think, I think I'm just reading too far into it, and that the indices do go up by one with every derivative you take.

ganeshie8 · Answer

they are doing that to be consistent in the coeffcients across y, y', y''

ganeshie8 · Answer

Notice that if c2 is 2 in y, it will be same in y' and y'' also

ganeshie8 · Answer

if you shift the index back to 0, the coefficients will be messed up

mendicant_bias · Answer

Alright, yeah, that makes sense. Thank you.

ganeshie8 · Answer

it looks a bit ugly as below if you shift the index back to 0

$$y'' = \sum\limits_{n=0}^{\infty} c_{n+2}(n+2)(n+1) x^{n} $$

ganeshie8 · Answer

if you think, its not that ugly actually...  you will need to shift it back to 0 to show them as power series i think

ganeshie8 · Answer

btw there seems to be a typo in power series for `y`  the index should start at 0, not 1