Question

Determine the first three nonzero terms in the Taylor polynomial approximation for the given initial value problem. y' = 2 sin y +4 e 2x; y(0) = 0 The Taylor approximation to three nonzero terms is y(x) = 2 + ...

Answer 1

Here, + 4e2a 20 @ We have the Taylar approximation polynomial is Y(a) = Y(0) + ft y '60) + y" (o)+ y"'(o) .23 y'= 2 sing ; y 100 ... y' (0) =2sin (y (0)) 148² 2 2 Sin (0)+4 = 4 y'a 2 2 cosy. y't 892x Then y" (0) - 2 Cos (409). y'ro)+8.6% 200 (0) 4'70)+8 2.0 = 2.478 216 Y" --2 siny. (YO)2+ 2Cosy.y" + 1622X Then y" (o)s –2 sin ( 4 (0))•(4'(0))*+2 60s (4(0)•y"CO) +16 820 - 2 sin (0) 14 16 +2.2010):16 +16 2.1.16+16 48 ... The Taylare appscoximation polynomial to the first threee non-zero terms is y (W) > n. 2.4'0+ +27 4" (0)+H3 y" (o) [": y0) -o] a4at 16.0² 16.indt 48. go 24X+827893

IVP is ala-22"ge"-tal -32"-ta', a NO)=-3.0-0 =0 © The Taylor approximation polyomial is – alt) = 2 (0) tk, a 10) * * * (0)+". Here given IVP 32"+372 20 ; 2CoI, X(0)0 Thus, ac-ta; a" (0) = -0.1(0)20 q".2-2-ta, 2"(0)= -2(0)-02-1 aiva-nl-nl-ta"=-22-ta" ai (0)2-22"(0) -0 20 24. Quiz-32" - "l-taiva-4a" taiv: avi(o)- qui(0) 2-4.(-1)-0 Therefore, required Taylor approximation polynomial upto first three t3 alt) > 200) + a'"10)+ to vi (A x', a", al, av all xvilo 3! 6! 1 +. (012+ 4 non-zero term is are zero at t20 .-+ 3 + 180 t6