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

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6. ó' + 2ó = õ, (ó ó(ï) = 0

)0

=

,0

7. ó' + 4ó = cos õ ó(0) = = 0, ó(ï) = 0

8. ó' + 4ó = sin õ, ó (0) = 0, ó(ï) = 0

9. ó' + 4ó = cos õ ó'(0) = 0, ó (ï) = 0

10. ó' + 3ó = cosõ ó'(0) = 0, ó'(ï) = 0

In each of Problems 11 through 16 find the eigenvalues and eigenfunctions of the given boundary

value problem. Assume that all eigenvalues are real.

11. ó' + Õó = 0, ó(0) = 0, ó '(ï) = 0

12. ó' + Õó = 0, ó'(0) = 0 ó(ï) = 0

13. ó' + Õó = 0, ó'(0) = 0 ó'(ï) = 0

14. ó' + Õó = 0, ó'(0) = 0 y(L) = 0

15. ó' + Õó = 0, ó'(0) = 0 y'(L) = 0

16. ó' --- Õó = 0, ó(0) = 0, ó '(L) = 0

17. In this problem we outline a proof that the eigenvalues of the boundary value problem (18), (19) are real.

(a) Write the solution of Eq. (18) as ó = k1 exp(i/ëõ) + k2 exp(-i/ëõ), where Õ = ë2, and impose the boundary conditions (19). Show that nontrivial solutions exist if and only if

exp(i /ëï) — exp(-i /ëï) = 0.

(i)

(b) Let ë = v + ia and use Euler’s relation exp(ivn) = cos(vn) + i sin(vn) to determine the real and imaginary parts of Eq. (i).

(c) By considering the equations found in part (b), show that a = 0; hence /ë is real and so is X. Show also that v = n, where n is an integer.

10.2 Fourier Series

Later in this chapter you will find that you can solve many important problems involving partial differential equations provided that you can express a given function as an infinite sum of sines and/or cosines. In this and the following two sections we explain in detail how this can be done. These trigonometric series are called Fourier series1;

1Fourier series are named for Joseph Fourier, who made the first systematic use, although not a completely rigorous investigation, of them in 1807 and 1811 in his papers on heat conduction. According to Riemann, when Fourier presented his first paper to the Paris Academy in 1807, stating that an arbitrary function could be expressed as a series of the form (1), the mathematician Lagrange was so surprised that he denied the possibility in the most definite terms. Although it turned out that Fourier’s claim of generality was somewhat too strong, his results inspired a flood of important research that has continued to the present day. See Grattan-Guinness or Carslaw [Historical Introduction] for a detailed history of Fourier series.

548

Chapter 10. Partial Differential Equations and Fourier Series

they are somewhat analogous to Taylor series in that both types of series provide a means of expressing quite complicated functions in terms of certain familiar elementary functions.

We begin with a series of the form

a0 / mnx . mnx \ , 4

f + E (am cos-^ + bm sin—) • (1)

m = 1

On the set of points where the series (1) converges, it defines a function f, whose value at each point is the sum of the series for that value of x. In this case the series (1) is said to be the Fourier series for f. Our immediate goals are to determine what functions can be represented as a sum of a Fourier series and to find some means of computing the coefficients in the series corresponding to a given function. The first term in the series (1) is written as a0/2 rather than simply as a0 to simplify a formula for the coefficients that we derive below. Besides their association with the method of separation of variables and partial differential equations, Fourier series are also useful in various other ways, such as in the analysis of mechanical or electrical systems acted on by periodic external forces.

Periodicity of the Sine and Cosine Functions. To discuss Fourier series it is necessary to develop certain properties of the trigonometric functions sin(mnx/L) and cos(mnx/L), where m is a positive integer. The first is their periodic character. A function f is said to be periodic with period T > 0 if the domain of f contains x + T whenever it contains x, and if

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