Question on series solutions of diff equations

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The discussion revolves around solving the differential equation (x-1)²y'' + (1/x)y' - 2y = 0, specifically identifying its singular points and determining their nature as regular or essential singularities. The original poster has identified the singular points but is uncertain about how this relates to the existence of a series solution of the form y = Σ (from j=0 to infinity) a_j x^(j+k). There is a suggestion that typically one assumes such a solution exists without needing to justify it through singular points. The conversation highlights a lack of responses, indicating possible confusion or frustration among participants regarding the question's clarity or formatting. Understanding the relationship between singular points and the existence of series solutions is crucial for solving the differential equation effectively.
blueyellow

Homework Statement


consider the differential equation

(x-1) squared y''+(1/x)y'-2y=0
find all the singular points of the equation and determine whether they are regular or essential singularities.
hence, explain why a solution of the form y= sigma (from j=0 to infinity) a (subscript j) x (to the power of j+k) should exist


The Attempt at a Solution


i found all the singular points but i don't see how that's going to help me determine why a solution of that form exists. usually doesn't one just ASSUME that a solution of that form exists?
thanks in advance
 
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hey r people not answering my question cos they r as clueless as i am or cos i somehow annoyed you by not typing out the subscripts properly? i don't know how to do that
 

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