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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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> 23.
> 24.
> 25.
> 26. 27.
f (x) = f (x) =
2  2x,
 2 x,
2x  2x2,
0,
x 2(3  x),
-2 < x < 0, 0 < x < 2;
3 < x < 0, 0 < x < 3
f (x + 4) = f (x) f (x + 4) = f (x) f (x + 6) = f (x)
Consider the function f defined in Problem 21 and let em (x) = f (x)  sm (x ).Plot|em (x )| versus x for 0 < x < 2 for several values of m. Find the smallest value of m for which |em(x)| <0.01 for all x.
Consider the function f defined in Problem 24 and let em (x) = f (x)  sm (x ).Plot|em (x )| versus x for 0 < x < 3 for several values of m. Find the smallest value of m for which |em(x)| < 0.1 for all x.
Suppose that g is an integrable periodic function with period T .
10.2 Fourier Series
557
(a) If 0 < a < T, show that
r T /> a+T
I g(x) dx = I g(x) dx.
J0 Ja
pa na+T
Hint: Show first thaw g(x) dx = I g(x) dx. Consider the change of variable s =
0T
x  T in the second integral.
(b) Show that for any value of a, not necessarily in 0 < a < T,
p T r a+T
I g(x) dx = I g(x) dx.
0a
(c) Show that for any values of a and b,
p a+T r b+T
I g(x) dx = I g(x) dx.
ab
28. If f is differentiable and is periodic with period T, show that f is also periodic with period T. Determine whether
F (x) = f f (t) dt
0
is always periodic.
29. In this problem we indicate certain similarities between three-dimensional geometric vectors and Fourier series.
(a) Let v1, v2, and v3 be a set of mutually orthogonal vectors in three dimensions and let u be any three-dimensional vector. Show that
u = a1v1 + a2v2 + a3v3, (i)
where
u  v.
a. =----L, i = 1, 2, 3. (ii)
vi  vi
Show that at can be interpreted as the projection of u in the direction of v. divided by the length of v..
(b) Define the inner product (u,v) by
fL
(u, v) = I u(x)v(x) dx. (iii)
Also let
(iv)
το (x) = cos(nn x /L), n = 0, 1, 2,...;
(x) = sin(nnx/L), n = 1, 2,....
Show that Eq. (10) can be written in the form a
(f^n) = 7 (τ0,τ) + 12 am (τς ) + 12 bm (^ς Σ (v)
m=1 ς=1
(c) Use Eq. (v) and the corresponding equation for (f, tyn) together with the orthogonality relations to show that
a = (f Το ) , n = 0, 1, 2,...; b = (f ^n) , n = 1, 2,.... (vi)
n (Τ,Τ) n (* ,f)
Note the resemblance between Eqs. (vi) and Eq. (ii). The functions το and tyn play a role for functions similar to that of the orthogonal vectors v1, v2, and v3 in three-dimensional
558
Chapter 10. Partial Differential Equations and Fourier Series
space. The coefficients an and bn can be interpreted as projections of the function f onto the base functions τ and ft .
nn
Observe also that any vector in three dimensions can be expressed as a linear combination of three mutually orthogonal vectors. In a somewhat similar way any sufficiently smooth function defined on  L < x < L can be expressed as a linear combination of the mutually orthogonal functions cos(nnx/L) and sin(nnx/L), that is, as a Fourier series.
10.3 The Fourier Convergence Theorem
In the preceding section we showed that if the Fourier series
a / m n x m n x \
f + E \m cos  + bm sin ) (1)
m = 1
converges and thereby defines a function f, then f is periodic with period 2L, and the coefficients am and bm are related to f (x) by the Euler-Fourier formulas:
1 fL m n x
am =  I f (x) cos-----------dx, m = 0, 1, 2,...; (2)
L JL L
1 fL m nx
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