set9 - Lecture 24 Boundary Value and Eigenvalue Problems...

Lecture 24
Boundary Value and Eigenvalue Problems for Linear Second Order
Differential Equations
We start with a model problem that you examined in Problem Set No. 1, the linear second
order DE,
y
′′
+ λy = 0 , λ ∈ R . (1)
The nature of the general solution for this DE depends on the parameter λ :
1. λ < 0: y ( x ) = c
1
e
√
− λx + c
2
e −
√
− λx (nonoscillatory)
2. λ = 0: y ( x ) = c
1
+ c
2
x (nonoscillatory)
3. λ > 0: y ( x ) = c
1
cos
√
λx + c
2
sin
√
λx (oscillatory)
As discussed in the early part of this course, a unique solution to the following initial value
condition:
y ( x
0
) = A, y
′
( x
0
) = B, A, B ∈ R , (2)
is guaranteed. When A = B = 0, then the unique solution is the trivial solution y ( x ) = 0.
When we consider boundary value problems of the form
y ( a ) = A, y ( b ) = B, (3)
there are many possibilities. For example, consider the special cse
y (0) = 0 , y ( π ) = 0 . (4)
1. If λ < 0, no nontrivial solution of (1) and (4) can exist. This follows from the fact, studied
earlier, that when q ( x ) < 0, nontrivial solutions can have at most one zero.
2. If λ = 0, then y ( x ) = c
1
+ c
2
x . But y (0) = 0 implies that c
1
= 0 and y ( π ) = c
2
π = 0
implies that c
2
= 0. Therefore the trivial solution y ( x ) = 0 is the only solution.
1

3. If λ > 0, then nontrivial solutions of (4) are possible. As mentioned earlier, the general
solution of (1) is
y ( x ) = c
1
cos
√
λx + c
2
sin
√
λx. (5)
The condition y (0) = 0 implies that c
1
= 0 so solutions will have the form
y ( x ) = c
2
sin
√
λx. (6)
But y ( π ) = 0 implies that sin
√
λπ = 0 which, in turn, implies that
√
λπ = nπ, n = 1 , 2 , · · · (7)
or
λ = n
2
, n = 1 , 2 , · · · . (8)
Thus the solutions of the BVP (4) are (up to a constant)
y
n
( x ) = sin nx, n = 1 , 2 , · · · . (9)
• The discrete λ -values λ
n
= n
2
, n = 1 , 2 , · · · , are called eigenvalues .
• The corresponding functions y
n
( x ) = sin nx , n = 1 , 2 , · · · , are called eigenfunctions . The
function y
n
( x ) has n − 1 nodes between 0 and π .
The reason for the “eigenvalue/eigenfunction” terminology is the following: Following stan-
dard practice, let us rewrite Eq. (1) as
Ly + λy = 0 , (10)
where L now represents the differential operator
d
2
dx 2
. We may rearrange this equation to have
the form
− Ly = λy, (11)
which may be viewed as an eigenvalue/eigenfunction equation: When operator − L acts on the
function y , it produces a multiple of y . This is analogous to the matrix eigenvalue/eigenvector
problems that you have studied in linear algebra:
Av = λ v , v ∈ R
n
. (12)
2

Later, we shall generalize L to include a general family of linear second order differential
operators.
We arrive at an important point: Nontrivial olutions to the DE in (1) exist for all values
of λ . Oscillatory solutions exist for all λ > 0 . The imposition of boundary conditions in (4)
implies that nontrivial solutions exist only for a discrete and countably infinite set of λ values,
namely λ
n
= n 2 .
Students in mathematical physics may recognize this BVP. Up to a multiplicative constant,
it describes the one-dimensional quantum mechanical particle in an infinite well, where the well
walls are located at x = 0 and x = π . The eigenvalues λ
n
, n = 1 , 2 , · · · represent the energies
of the various eigenstates of this “particle-in-a box” model.