Question

Question Help * Determine the first three nonzero terms in the Taylor polynomial approximation for the given initial value problem 4x" + 9tx = 0, x(0-1, x'(0) :0 The Taylor approximation to three nonzero terms is x(t) = | | +

Answer 1

Given that, 4x double prime + 9 t x = 0 x (0) = 1 x prime (0) = 0} (1) x = sigma^infinity_n = 0 c_n t_n be the series x prime = sigma^infinity_n = 0 n_c_n t^n - 1 ---(2) x double prime = sigma^infinity_n = 0 n(n - 1) c_n t^n - 2---(3) Now, substituting the (2) & (3) values in equation (1) doublerightarrow 4 [sigma^infinity_n = 0 n(n - 1) c_n t^n - 2] + 9 t (sigma^infinity_n = 0 c_n t^n) = 0 doublerightarrow 4[sigma^infinity_n = 0 n(n - 1)c_n t^n - 2] + 9[sigma^infinity_n = 0 c_n x^n + 1] = 0 4[sigma^infinity_n = 0 (m + 2) (n + 2 - 1) c_n + 2] + 9[sigma^infinity_n = 1 c_n - 1 x^n - 1 + 1] = 0 (0 + 2 m (0 + 1) c_0 + 2 + sigma^infinity_n = 1 c_n + 2 0 + 1 c_n + 2 t^n + sigma^infinity_n = 1 c_0 + t) = 0