Question

Consider the tollowing dorential equation Now For ach part below you gve your anowers in ns of actions (as appropni t decima cacuistng Emter the roots to the indicial eotion below You must en the oots in the rder of smliest to largest separsted by a comma 09 Note 1You must include an quas Note2: You must use the symbol 85your ron le) Hence enter the fest thee on-ero terms of the soon corsponting to the largest vdoa

Answer 1

Xy'' - 8 y' + 7 xy = 0 Rightarrow y'' - 3/x y' + 7y = 0....(1) here p(x) = -3/x q(x) = 7 xp (clearly x = 0 is a singular point x p(x) = -8 x^2 q(x) = 7 x^2 therefore x p(x) and x^2 q(x) both are analytic at x = 0 so x = 0 is a regular singular point let Q = sigma^infinity_n = 0 a_n x^n - r be a solution of (1) [a_0 notequalto 0] Q = sigma^infinity_n = 0 (n + r)a_n x^n + r - 1 Q'' = sigma^infinity_n = 0 (n + r) (n + r - 1) a_n x^n + r - 2 from (1)we have xy'' - 8 y' + 7 xy = 0 Rightarrow x Q'' - 8 Q' + 7x Q = 0 Rightarrow x sigma^infinity_n = 0 (n + r) (n + r - 1) a_n x^n + r - 2 - 8 sigma^infinity_n = 0 n + r)a_n x^n - r - 1