Question

Problem #6: Let 58 0 < t <a f(0) = -8 x<I< 27 and assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is even Consider the following differential equation. 3 d2x de 2 + 7x = $(1) Find a particular solution of the above differential equation of the form 00 00 Xp(1) git,n) P and enter the function g(t, ) into the answer box below.

Answer 1

Note :

1) To solve this problem ist we find Fourier series form of f(t) and putt it in the given ODE.

2) Next , we putt particular solution x(t) in given ODE and compare the cosfficients of Fourier series on both sides.

# First we elun srctcar Hence A = 2 JL - 8 ΟΥ auzo Now find Faurer-series of f(t) let fet? en [ an cos (nl) the Sun (nett] Sinu tét)= ; OLAC -funchon on (o, o) $64). Sin manie) is odd - Lunchun bn=0 S' faturdt =) go = la [ 8.98 dt'+57352.89 dt] [ 11- (251–57)] = 0 3. Do 9 fet) cos( min Hold i b= 27 = 118.COM (4) dt 87 CM( 94301] [{sm(t)}-(){ SME] 16 [{sın (qal). - of-{ sin(510) - Sin(172)}] (m) sin(22) jnc 2,4,6,- 1 232 233 sin (m) ដែន s no 1,3,5,7 Now Since 2041) = An Cop (nl) is solution of GDE § 3"(+) +72 = $CH) s hence XpCt) sakerfiel vt lc. + 3 [2 (-B) An Coll 2t) ]++ [ se on An Col (26)]= , COA (46) 02-572 +7} An - Coll!) E o an Cod (b) (387309 An = by comparing coeficient of Collet) Sm(1) Sn=1,3,5). (29-30) CH) This required foluhun gut,n) green by cs Scanned with nor و با - al Sin (nozu an nel 12 I π π T an 4 an 128 D(28-3n2) 579,49, An = An Cos (9) Cam Scanner