Question

5. Consider the second order equation x" + x = 0 with initial conditions (0) = 1, x'(0) = 0. We know the solution is x(t) = cos(t). Recover the exact solution by using the Picard iterative method to solve the first order system that is equivalent to the second order equation above.

Answer 1

Given problem is d2k + U=0 where , u=l, dx= 0, when tzo. dt? let dor=y. So, that d2e = dy Them we have de = yen dy -t, where, 221, yzo when tzo 이온 A ditz We know the nth Approximation (an. Yn) to the initial Valve problem da aff241) ; dy = 840, 84, 8), when, 2=30, Y=30 at t-to- given by 2 = xo + \ $(t, 2m4, 354) dt -6) to - In a yout (t, anyo Yn-1)dt - ③ is Compare egne) 4 (31, we have Altia,y)=y, glt, u, y) = -1 =1, Yoo When to =0 - . From eang un= 1+ In-idt - 4 from ean @ In = 0 + 1*-un-idt In = 0 - 1*xadt -

First_Aprilozimation putting nembo omo legno we get sy = 1 + 5*odt - 14[ordt - 1 - 0 30.2, 0-4*%dt = -fedt -® Second Approximation, putting nzi in egno Lusingan =} + {*3dt - it fitt - .)/2 = 1 *- - pulting n=2 in en 6 and using en ③ Jy = -1*2405_=-11.00 = - + - Thised opporo ximation'- petting mz3 in eam @ Lusing a H3 = tl yedt 1-1ttat .243 = 1- .. -13 pulting n=3 in ean B and using eam - -43420t --4415 Helt Vo -ya-+1+3 -

Foweth approximation - Pulting med imeno - A ugimey ogn @. 964 = 1 +f*ladt a 1+4+ + Lif3fdt H4 = 1- * hity - Putting n=4 in een 6 & using B Yoga t* *selt --440-3j4t 3-47-year-47-4) – ® Fifth approximation putting n=s in ean B cand using @ *5=1+ -(4-]dt _31+[-*+It 245 1-toate acestea -@

from B & eam 16 It is clear that au = 45 _so, approximation soin of given Problemen 2l = 1- 2 + 1 lies es 94