Question

1) a) Assume y is a Maclaurin series in x and find explicitly the first six non-zero terms of the power series solution of y" 4y -0.

Answer 1

Given data, Assume y is a Maclaurin series in x & Explicitly the first six non-zero terms of the power series solution of y'' + 4 y = 0 rightarrow (1) We assume y (x) a Maclaurin series form. y (x) = Sigma^infinity_n = 0 a_n x^n rightarrow (2) this gives, y' = Sigma^infinity_n = 0 n a_n x^n - 1, y'' = Sigma^infinity_n = 0 n (n - 1) a_n x^n - 2 Substitution in the Equation in (1) then gives, Sigma^infinity_n = 0 n (n - 1) a_n x^n - 2 + 4 Sigma^infinity_n = 0 a_n x^n = 0 rightarrow (3) Rightarrow 0 + 0 Sigma^infinity_n = 0 n (n - 1) a_n x^n - 2 + 4 Sigma^infinity_n = 0 a_n x^n = 0 for simplicity, we replace n by n - 2 in the second series. \r\n Therefore, (3) reduces to Sigma^infinity_n = 2 n (n - 1) a_n x^n - 2 + 4 Sigma^infinity_n = 2 a_n - 2 a_n - 2 x^n - 2 = 0 Rightarrow Sigma^infinity_n = 2 (n (n - 1) a_n + 4 a_n - 2) x^n - 2 = 0 = Sigma^infinity_n = 0 0 x^n - 2