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:
(x2 - 1 ) y'' + x y - y = 0
The Attempt at a Solution
(again) - the solution is in the book - it's an example problem
y = ∑cnxn
y' = ∑ (n) cnxn-1
y'' = ∑ (n) (n-2) cnxn-2
(x2 +1 ) ∑(n)(n-1)cnxn-2 + (x)∑(n)cnxn-1 - ∑cnxn
Divided through by x2 (after cross multiplying the x2 + 1)
the first summation the x2's cancel, the second summation (the term that results from multiplying by the "1" will subtract the "2" exponent in the xn-2 term, the third summation leaves an x in the denominator and as a result will subtract a "1" in the exponent of xn-1
= ∑ (n)(n-1) cn xn + ∑ (n)(n-1)cn xn-2 + ∑(n)cn xn
now for my question - the forth summation - the "-y" term in the original equation
dividing through by x2 - it looks like the last summation should be ∑cn x-2
the book has ∑cn xn for the last term - as if it was not divided by the x2 term
what am I missing, regarding the last term ( the lone y term) after dividing through by x2?
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