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and by that, we re a ch the two equations: T
0
= ¡λT , and X
00
= ¡λX. But the solution u must
satisfies in addition to the equation, the given boundary conditions at x = 0; L , that is,
X(0)T (t) = 0; X(L) T (t) = 0 :
If T (t) is identically zero, then u(x; t) is identically zero, that is in general is not acceptable
(except the initial condition o f the equation is zero and then the solution is trivial). Therefore,
we obtain two boundary conditions for X(x) as X(0) = X(L) = 0. Therefore, we look for
function X(x) that satisfies the following equation
8
<
:
d
2
dx
2
X = ¡λX
X(0) = X(L) = 0
: (7.8)
Equations of the above form are g enerally called eigenvalue problems. In fact, we would like
to find non-trivial fun ctions X(x) called eigenfunctions that preserves th eir structure under
the second order differentiation. The values λ in the above equation are called eigenvalues.
Therefore, solving this type of equations is crucial to solve a partial differential equation.
7.3.1 General Dirichlet eig envalue problem
The most general form of the Dirichlet eigenvalue problem is
a(x) X
00
+ b(x) X
0
+ c(x)X = ¡λX
X(x
0
) = 0; X(x
1
) = 0
;
where a(x) > 0 in the interval [x
0
; x
1
]. As we learned in the previous chapters, we are unable,
in general, to solve equations with the variable coefficients in closed form, and for this, there
is no hope to determine the closed form solutions to the ab ove eigenvalue problem. However,
we can say several important things about the eigenva lues and eigenfunctions of the problem.
The inte rested reader is referred to the s eco nd volume of this bo o k for a detailed discussion
on this subject. Here, we restrict ourselves to simple eigenvalue problem.
Of most impo rtant are the following simple problem (7.8). Let us solve the equation.
The characteristic polynomial is r
2
= ¡λ and thus r
1;2
= ± ¡λ
p
. There are three possible
cases for λ, that are λ < 0, λ = 0, and λ > 0. We first show that there is no eigenfunction for
λ ≤0. Multiply the equation by X(x) and integrate in the interval [0; L]. We obtain
Z
0
L
X
00
(x) X(x) = ¡λ
Z
0
L
jX(x)j
2
:
Integration by parts formula, simplifies the left h and side of above relation to
Z
0
L
X
00
(x) X(x) = X
0
(x)X(x)j
0
L
¡
Z
0
L
jX
0
(x)j
2
= ¡
Z
0
L
jX
0
(x)j
2
:
Note that the boundary term is zero in the above relation due to the boundary conditions
at x = 0; L. We obtain finally
Z
0
L
jX
0
(x)j
2
= λ
Z
0
L
jX(x)j
2
;
7.3 Eigenvalue problem 7