I'm a little confused with ODEs. After two weeks of trying to figure out Frobenius I have finally realized that there seems to be two different power set used by all of my three books for the y substitution but I am unsure when to use either one. Here are the two sets that i'm talking about:(adsbygoogle = window.adsbygoogle || []).push({});

Set 1 from section 6.1

[tex] y = \sum_{n=0}^\infty C_n x^n [/tex]

[tex] y\prime = \sum_{n=1}^\infty n C_n x^{n-1}[/tex]

[tex] y\prime\prime = \sum_{n=2}^\infty n(n-1) C_n x^{n-2}[/tex]

set 2 from section 6.2

[tex] y = \sum_{n=0}^\infty n C_n x^{n-1} [/tex]

[tex] y\prime = \sum_{n=0}^\infty n C_n x^{n-1} [/tex]

[tex] y\prime\prime = \sum_{n=0}^\infty n(n-1) C_n x^{n-2} [/tex]

I think I have come to understand that I should use set 1 if and only if all of the singular points are irregular and set 2 when I have at least one regular singular point. Is this correct? If not, when is it appropriate to use set 1 rather than set 2? Or is set 2 only designed to work with Frobenius' method while set 1 only works lacking a Taylor power series expansion?

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# Frobenius Method

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