Question

Consider the solution to the IVP

Find the coefficient of

$x^5$in its Taylor expansion centered at 0.

Answer 1

Ay'-150 7(0)=1 ,y' (0-2 48-4)2 = 0 >> (4-40 yao gry g'=o ar 7 du 2 ay 7 say => =da dy =o.da jooda [9 = C1, as is arbitraty constant. a > se fax -> nogs = et o is arbitrary constant X => 14 zelo -a) Now , 7(0) = 1 =0 Pulting in equation (1) - gex) = aq yeo) = 9 => 1 = = 1 y= 1 In this case, the coefficient of 25 is Putting X=0 in equation (2) - year= gen > yco) ale

1 => y ex 3 Now, the Taylon series of f (1) about the point o is? gcas = $(0) + (8-0) (0 )+(8-0)? F"(0) + (80 02 "CO) GMCO) + (1-0) Rev.com 4! 5! ge So, the coefficient of the taylor 25 (8-0)5 in series .expornsion of f(x) is. 5! Wow, f(x) » f(x) = e2 → f (x)=en > f (x)=en = try (w-e? (%) en ě zl. * CV (0)=e fN310) so, 120 1/3 Therefore, the Taylor of the given Coefficient of n5 in the expansion centered of the solution al 0 IVP is 1.20