(a) Show that the Fourier series of the periodic function f(t) of period 27, defined as t/n, if 0

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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(a) Show that the Fourier series of the periodic function f(t) of period
27, defined as
S(t) = { t/T, if 0<t< z,
(27 – t)/7, if 7 <t< 2r,
is given by
cos (2n + 1)t)
(2n + 1)2
f(t) =
(b) Draw the plots of Fourier series of f(t) obtained in 1(a) by taking
finite number of terms of the series, together with the plot of f(t).
Further, comment on the convergence of the Fourier series.
(c)
that, when w is not an integer, the function y(t) given by
. cos (2n + 1)t) – cos(wt)
(2n + 1)²(w² – (2n + 1)²]
y(t) =
221- cos
- cOS wt)
satisfies the initial value problem (IVP) arising from undamped
forced oscillations of a mass-spring system given by
dy
+w*y = f(t),
dt?
dy
= 0
dt
at t= 0.
(d) Plot the solution y(t) given in 1(c) by taking 10-terms of the
Fourier series with w = 1 and comment on the same.
Transcribed Image Text:(a) Show that the Fourier series of the periodic function f(t) of period 27, defined as S(t) = { t/T, if 0<t< z, (27 – t)/7, if 7 <t< 2r, is given by cos (2n + 1)t) (2n + 1)2 f(t) = (b) Draw the plots of Fourier series of f(t) obtained in 1(a) by taking finite number of terms of the series, together with the plot of f(t). Further, comment on the convergence of the Fourier series. (c) that, when w is not an integer, the function y(t) given by . cos (2n + 1)t) – cos(wt) (2n + 1)²(w² – (2n + 1)²] y(t) = 221- cos - cOS wt) satisfies the initial value problem (IVP) arising from undamped forced oscillations of a mass-spring system given by dy +w*y = f(t), dt? dy = 0 dt at t= 0. (d) Plot the solution y(t) given in 1(c) by taking 10-terms of the Fourier series with w = 1 and comment on the same.
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