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We shall look for a power series solution around x o = 2;These issues are settled by the theory of power series and analytic functions 12 Power series and analytic functions A power series about a point x0 is an expression of the form X n=0 ∞ a n (x − x0) n = a 0 a1 (x − x0) a2 (x − x0) 2 (24) Following our previous discussion, we want to know whether this infinite sum indeedSolution 1 First assume y 2 =ux 2, and y 2 '=2uxu'x 2 and y 2 ''=u''x 2 4u'x2u Substituting into x 2 y''2xy'6y=0 we get x 2 (u''x 2 4u'x2u)2x(2uxu'x 2)6y=0 and simplifying by adding like terms we get u''x 4 6u'x 3 =0 We reduce the order by w=u' to get w'x 4 6wx 3 =0 Now dividing by wx 4 and rearranging, we get Calameo Solution Techniques For Elementary Partial Differential Equations 2nd Constanda (1-x^2)y''-2xy' 6y=0 power series solution