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## Homework Statement

Hello,

I suspect this is an easy answer but I am not seeing it. I am reviewing (more so for fun / hobby) some differential equations – I’m not in school.

I’m needing help with an example problem in Differential Equations With Boundary-Value Problems Zill 2nd edition. In googling around for a possible errata, I see it’s also in other editions – not an error – it’s me.

It’s Chapter 6, section 2, example7, of the 2nd edition (my book from way back) and Chapter 6, section 1, example 4 of the 7th edition.

The problem is to solve by way of a power series:

(x

^{2}- 1 ) y'' + x y - y = 0

## Homework Equations

## The Attempt at a Solution

(again) - the solution is in the book - it's an example problem

y = ∑c

_{n}x

^{n}

y' = ∑ (n) c

_{n}x

^{n-1}

y'' = ∑ (n) (n-2) c

_{n}x

^{n-2}

(x

^{2}+1 ) ∑(n)(n-1)c

_{n}x

^{n-2}+ (x)∑(n)c

_{n}x

^{n-1}- ∑c

_{n}x

^{n}

Divided through by x

^{2}(after cross multiplying the x

^{2}+ 1)

the first summation the x

^{2}'s cancel, the second summation (the term that results from multiplying by the "1" will subtract the "2" exponent in the x

^{n-2}term, the third summation leaves an x in the denominator and as a result will subtract a "1" in the exponent of x

^{n-1}

= ∑ (n)(n-1) c

_{n }x

^{n}+ ∑ (n)(n-1)c

_{n}x

^{n-2}+ ∑(n)c

_{n}x

^{n}

now for my question - the forth summation - the "-y" term in the original equation

dividing through by x

^{2}- it looks like the last summation should be ∑c

_{n}x

^{-2}

the book has ∑c

_{n}x

^{n}for the last term - as if it was not divided by the x

^{2}term

??

what am I missing, regarding the last term ( the lone y term) after dividing through by x

^{2}?

Thanks

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