MATH201 Lecture 23: MATH201(LEC23)
15 Oct 2016
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Recall: if ! is a regular point, then the equation. + = 0 ( ) has an analytic solution at !, that is: where. Theorem: if ! is a regular singular point for the equation (*), then multiply (*) by and rewrite as: where. And = : notice the similarity between (**) and the cauchy_euler equation. 1 where ! is the biggest root of the polynomial: 1 + ! where ! and ! are the first terms of the expansions of and respectively: Step 1: classify the type of point ! is. Analytic: since p(x) and q(x) are both not analytic, the point ! is singular. 2: since a(x) and b(x) are both analytic, the point ! is regular. Therefore: ! is a regular-singuler point, so we use frobenius method. Step 2: since ! is a regular-singuler point, use frobenius method. (i) + = 0 then our given equation looks like. By comparing the above two equations, we get:
