Answered: (a) Show that the Fourier series of the…

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

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(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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