1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

534 CHAPTER 12 • FOURJER SERI ES AND THE LAPLACE TRANSFORM

We obtain the steady st ate solution by assuming that Up (t) has the Fourier
series representation

00 00

Up (t) = ~o + 2::>n cosnt + 'L)n sin nt,

n=l n=l

and that u;, (t) and ui (t) can be obtained by termwise differentiation:

00 00

2u; (t) = 2 L nbn cosnt- 2 L na.,.sinnt, and

n=l n=l

00 00

u;(t) = - L:n^2 ancosnt-L:n^2 b,,sinnt.

n=l n=l

Substituting these expansions into the differential equation results in

00

F (t) = a

2

° + L [(1-n^2 ) a,.+ 2nb,.} cosnt

n=l

00

+ L [-2nan + (1 - n

2

) bn} sinnt.

n = l

Equating the coefficients with the given series for F (t), we find that ~ = 0,

and that

when n is odd;

when n is even.

(l -n

2

) an+ 2nb,. = { ;2

• 2na.,. + (1 - n^2 ) bn = 0 for all n.

Solving this linear system for a,. and b,,, we get

1-n^2

n2 (1 + n2)2'

0,

2n

The general solution is

for n odd;

for n even.

for n odd;

for n even.

(^00) 1 - (2n- 1) 2
U (t) = A1e- t + A2te-t + L 2 cos [(2n - 1) t]
n=l (2n - 1)^2 [ 1 + (2n - 1)^2 ]
~ 2(2n- 1)

• L 2 sin[(2n-l)t).
  n=l (2n-1)
  2
  [ 1 + (2n- 1)^2 ]