Question

It’s a fact that Taylor series can be used to study approximate solutions to non-linear ODEs. Use this fact to study approximate solutions to d 2x dt2 + sin x = 0, (1) subject to x(0) = 0 and x 0 (0) = x1 ≥ 0. Discuss conditions under which your reasoning is accurate, and make sure to include everything you can say. Make choices for x1 to illustrate your points ONLY; make sure to address the problem in full generality in a separate conclusive paragraph.

It's a fact that Taylor series can be used to study approximate solutions to non-linear ODEs. Use this fact to study approximate solutions to d²x + sin 2 0, (1) dt2 subject to x(0) = 0 and x'(0) = x1 > 0. Discuss conditions under which your reasoning is accurate, and make sure to include everything you can say. Make choices for Xi to illustrate your points ONLY; make sure to address the problem in full generality in a separate conclusive paragraph.

Answer 1

dx X (0) = 24 + sina = o x(0) = 0, at 80 let a(t) = { anth n-1 x'(t) = Enant nz n-2 En(n-dant n=2 using sinx= x- tas 3] जा anthj Zn(n-1) anth-2 + + Zanth n=0 31 n=2 get so we ao=0 given x (o)=0 so En(n) anth-2 tte anth n-1 e la antrip + te ontbingo 120 n=1 n=2 80 compasing constant team x'(o)=x, =a 2(1a2 O 1 comparing coefficient of t 3 (2) azt a 0 6 a3 = al az = - ab -> 6 comparing Coefficient of t 12 ay taz ay=0

Comparing coefficient of t 20 ast az o 6 olo o 20 as 291 6 as = = 60 ai 60 Com pasing coefficient of t 4 Зоа, тач 3 a, az o 6 ab O since (azzay =o) 3 alt) ll (ant- - lt 6 +34 €) 60 Ane