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# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
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> 29. f (x) = (4x2 — 4x — 3)/4, 0 < x < 2
> 30. f (x) = x3 — 5x2 + 5x + 1, 0 < x < 3
31. Prove that if f is an odd function, then
/:
rL
f (x) dx = 0.
—L
32. Prove properties 2 and 3 of even and odd functions, as stated in the text.
33. Prove that the derivative of an even function is odd, and that the derivative of an odd function is even.
fx
34. Let F(x) = I f (t) dt. Show that if f is even, then F is odd, and that if f is odd, then
0
F is even.
35. From the Fourier series for the square wave in Example 1 of Section 10.3, show that
n = 1 — 1 +1 — ³ + ••• = ? <—^
4 3 5 7 2n + 1
n=0
This relation between n and the odd positive integers was discovered by Leibniz in 1674.
36. From the Fourier series for the triangular wave (Example 1 of Section 10.2), show that
n2 11 ^ 1
³ ³ x—'
= 1 + ^2 + 72 +-=E-
8 ' ' 32 ¦ 5^ n=0 (2n + 1)2'
37. Assume that f has a Fourier sine series
f (x) = E bn sin(nnx/L), 0 < x < L.
n=1
(a) Show formally that
2 Ã L ^
- [f (x)]2 dx = ]T b2n.
L J0 n=1
Compare this result with that of Problem 17 in Section 10.3. What is the corresponding result if f has a cosine series?
(b) Apply the result of part (a) to the series for the sawtooth wave given in Eq. (9), and thereby show that
n2 11 ^ 1
—— = 1 +—2 +—2 + •••=/ ~2.
6 22 32 n=1 n2
This relation was discovered by Euler about 1735.
More Specialized Fourier Series. Let f be a function originally defined on 0 < x < L .In this section we have shown that it is possible to represent f either by a sine series or by a cosine series by constructing odd or even periodic extensions of f, respectively. Problems 38 through 40 concern some other more specialized Fourier series that converge to the given function f on (0, L).
572
Chapter 10. Partial Differential Equations and Fourier Series
38. Let f be extended into (L, 2L] in an arbitrary manner. Then extend the resulting function into (-2L, 0) as an odd function and elsewhere as a periodic function of period 4L (see Figure 10.4.6). Show that this function has a Fourier sine series in terms of the functions sin(nnx/2L), n = 1, 2, 3,...; that is,
f (x) = E bn sin(nnx/2L),
n
n=1
where
1 [2L
bn = — I f (x) sin(nnx/2L) dx.
L Jo
This series converges to the original function on (0, L).
L 1 1
1 : i
L
1
³ ,
1 ³ L 2L x
³ i
Ó ‘ .-------N. 1
! 1
³ 1
³ ³ ; r
1 ³
-2L - L L 2L x
i i
l4^ _ S.
1 1
FIGURE 10.4.6 Graph of the function in Problem 38.
FIGURE 10.4.7 Graph of the function in Problem 39.
39. Let f first be extended into (L, 2L) so that it is symmetric about x = L; that is, so as to satisfy f (2L — x) = f (x) forO < x < L. Let the resulting function be extended into (—2L, 0) as an odd function and elsewhere (see Figure 10.4.7) as a periodic function of period 4 L. Show that this function has a Fourier series in terms of the functions sin (n x /2L), sin(3nx/2L), sin(5nx/2L), ...; that is,
f (x ) = E bn
(2n — 1)n x
2L ’
where
fL . (2n — 1)nx
I f (x) sin------—---------dx.
J0
2L
This series converges to the original function on (0, L].
40. How should f, originally defined on [0, L], be extended so as to obtain a Fourier series involving only the functions cos(nx/2L), cos(3nx/2L), cos(5nx/2L), ...? Refer to Problems 38 and 39. If f (x) = x for 0 < x < L, sketch the function to which the Fourier series converges for —4L < x < 4L.
10.5 Separation of Variables; Heat Conduction in a Rod
573
10.5 Separation of Variables; Heat Conduction in a Rod
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