EXAM 3 BONUS QUESTION MATH 2153 xn , which is absolutely conver1 + n2 n=0 gent on [1, 1], satises there the dierential equation Show that the power series f (x) = 2 d 1 = f (x) + x 1x dx f (x) = 1+ x d dx 2 f (x), (1+n2 )xn , absolutely convergent on (1, 1), satises while the series g(x) = n=0 d 1 + x g(x) = 1x dx 2 1 = 1x d 1+ x dx 2 1 . 1x Do you notice anything about the relations above? Guess the power series solution that solves F (x)+2xF (x) = 1/(1x), and likewise guess the Maclaurin 1 d 1 series of G(x) = 1x + 2x dx 1x . d (In our shorthand notation here, x dx Oklahoma State University, Fall 2015 1 2 d u(x) = x dx x d u(x) dx .)