Handbook of mathematics for engineers and scienteists part 74 doc

12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 479
where f = f(x), k =tan

π
2m

.
3
◦
.Letm be an odd integer. Then,
y
1
=
⎧
⎪
⎨
⎪
⎩
|f(x)|
–1/4
cos

–
1
ε

x
0

|f(x)| dx +
π

4

if x < 0,
1
2
k
–1
[f(x)]
–1/4
exp

1
ε

x
0

f(x) dx

if x > 0,
y
2
=
⎧
⎪
⎨
⎪
⎩
|f(x)|
–1/4

cos

–
1
ε

x
0

|f(x)| dx –
π
4

if x < 0,
k[f(x)]
–1/4
exp

–
1
ε

x
0

f(x) dx

if x > 0,
where f = f(x), k =sin


π
2m

.
12.2.3-4. Equations not containing y

x
. Equation coefficients are dependent on ε.
Consider an equation of the form
ε
2
y

xx
– f(x, ε)y = 0 (12.2.3.4)
on a closed interval a ≤ x ≤ b under the condition that f ≠ 0. Assume that the following
asymptotic relation holds:
f(x, ε)=
∞

k=0
f
k
(x)ε
k
, ε → 0.
Then the leading terms of the asymptotic expansions of the fundamental system of solutions
of equation (12.2.3.4) are given by the formulas
y
1

= f
–1/4
0
(x)exp

–
1
ε


f
0
(x) dx +
1
2

f
1
(x)
√
f
0
(x)
dx


1 + O(ε)

,
y

2
= f
–1/4
0
(x)exp

1
ε


f
0
(x) dx +
1
2

f
1
(x)
√
f
0
(x)
dx


1 + O(ε)

.
12.2.3-5. Equations containing y


x
.
1
◦
. Consider an equation of the form
εy

xx
+ g(x)y

x
+ f(x)y = 0
on a closed interval 0 ≤ x ≤ 1. With g(x)>0, the asymptotic solution of this equation,
satisfying the boundary conditions y(0)=C
1
and y(1)=C
2
, can be represented in the
form
y =(C
1
– kC
2
)exp

–ε
–1
g(0)x


+ C
2
exp


1
x
f(x)
g(x)
dx

+ O(ε),
where k =exp


1
0
f(x)
g(x)
dx

.
480 ORDINARY DIFFERENTIAL EQUATIONS
2
◦
. Now let us take a look at an equation of the form
ε
2
y


xx
+ εg(x)y

x
+ f(x)y = 0 (12.2.3.5)
on a closed interval a ≤ x ≤ b. Assume
D(x) ≡ [g(x)]
2
– 4f(x) ≠ 0.
Then the leading terms of the asymptotic expansions of the fundamental system of solutions
of equation (12.2.3.5), as ε → 0, are expressed by
y
1
= |D(x)|
–1/4
exp

–
1
2ε


D(x) dx –
1
2

g

x
(x)

√
D(x)
dx


1 + O(ε)

,
y
2
= |D(x)|
–1/4
exp

1
2ε


D(x) dx –
1
2

g

x
(x)
√
D(x)
dx



1 + O(ε)

.
12.2.3-6. Equations of the general form.
The more general equation
ε
2
y

xx
+ εg(x, ε)y

x
+ f(x, ε)y = 0
is reducible, with the aid of the substitution y = w exp

–
1
2ε

gdx

, to an equation of the
form (12.2.3.4),
ε
2
w

xx

+(f –
1
4
g
2
–
1
2
εg

x
)w = 0,
to which the asymptotic formulas given above in Paragraph 12.2.3-4 are applicable.
12.2.4. Boundary Value Problems
12.2.4-1. First, second, third, and mixed boundary value problems (x
1
≤ x ≤ x
2
).
We consider the second-order nonhomogeneous linear differential equation
y

xx
+ f(x)y

x
+ g(x)y = h(x). (12.2.4.1)
1
◦
. The first boundary value problem: Find a solution of equation (12.2.4.1) satisfying the

boundary conditions
y = a
1
at x = x
1
, y = a
2
at x = x
2
.(12.2.4.2)
(The values of the unknown are prescribed at two distinct points x
1
and x
2
.)
2
◦
. The second boundary value problem: Find a solution of equation (12.2.4.1) satisfying
the boundary conditions
y

x
= a
1
at x = x
1
, y

x
= a

2
at x = x
2
.(12.2.4.3)
(The values of the derivative of the unknown are prescribed at two distinct points x
1
and x
2
.)
12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 481
3
◦
. The third boundary value problem: Find a solution of equation (12.2.4.1) satisfying the
boundary conditions
y

x
– k
1
y = a
1
at x = x
1
,
y

x
+ k
2
y = a

2
at x = x
2
.
(12.2.4.4)
4
◦
. The third boundary value problem: Find a solution of equation (12.2.4.1) satisfying the
boundary conditions
y = a
1
at x = x
1
, y

x
= a
2
at x = x
2
.(12.2.4.5)
(The unknown itself is prescribed at one point, and its derivative at another point.)
Conditions (12.2.4.2), (12.2.4.3), (12.2.4.4), and (12.2.4.5) are called homogeneous if
a
1
= a
2
= 0.
12.2.4-2. Simplification of boundary conditions. The self-adjoint form of equations.
1

◦
. Nonhomogeneous boundary conditions can be reduced to homogeneous ones by the
change of variable z =A
2
x
2
+A
1
x+A
0
+y (the constants A
2
, A
1
,and A
0
are selected using
the method of undetermined coefficients). In particular, the nonhomogeneous boundary
conditions of the first kind (12.2.4.2) can be reduced to homogeneous boundary conditions
by the linear change of variable
z = y –
a
2
– a
1
x
2
– x
1
(x – x

1
)–a
1
.
2
◦
. On multiplying by p(x)=exp


f(x) dx

, one reduces equation (12.2.4.1) to the
self-adjoint form:
[p(x)y

x
]

x
+ q(x)y = r(x). (12.2.4.6)
Without loss of generality, we can further consider equation (12.2.4.6) instead of
(12.2.4.1). We assume that the functions p, p

x
, q,andr are continuous on the inter-
val x
1
≤ x ≤ x
2
,andp is positive.

12.2.4-3. Green’s function. Linear problems for nonhomogeneous equations.
The Green’s function of the first boundary value problem for equation (12.2.4.6) with
homogeneous boundary conditions (12.2.4.2) is a function of two variables G(x, s)that
satisfies the following conditions:
1
◦
. G(x, s) is continuous in x for fixed s, with x
1
≤ x ≤ x
2
and x
1
≤ s ≤ x
2
.
2
◦
. G(x, s) is a solution of the homogeneous equation (12.2.4.6), with r = 0,forall
x
1
< x < x
2
exclusive of the point x = s.
3
◦
. G(x, s) satisfies the homogeneous boundary conditions G(x
1
, s)=G(x
2
, s)=0.

4
◦
. The derivative G

x
(x, s)hasajumpof1/p(s) at the point x = s,thatis,
G

x
(x, s)


x→s, x>s
– G

x
(x, s)


x→s, x<s
=
1
p(s)
.
For the second, third, and mixed boundary value problems, the Green’s function is de-
fined likewise except that in 3
◦
the homogeneous boundary conditions (12.2.4.3), (12.2.4.4),
and (12.2.4.5), with a
1

= a
2
= 0, are adopted, respectively.
482 ORDINARY DIFFERENTIAL EQUATIONS
The solution of the nonhomogeneous equation (12.2.4.6) subject to appropriate homo-
geneous boundary conditions is expressed in terms of the Green’s function as follows:*
y(x)=

x
2
x
1
G(x, s)r(s) ds.
12.2.4-4. Representation of the Green’s function in terms of particular solutions.
We consider the first boundary value problem. Let y
1
= y
1
(x)andy
2
= y
2
(x) be linearly
independent particular solutions of the homogeneous equation (12.2.4.6), with r = 0,that
satisfy the conditions
y
1
(x
1
)=0, y

2
(x
2
)=0.
(Each of the solutions satisfies one of the homogeneous boundary conditions.)
The Green’s function is expressed in terms of solutions of the homogeneous equation
as follows:
G(x, s)=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
y
1
(x)y
2
(s)
p(s)W (s)
for x
1
≤ x ≤ s,
y
1
(s)y
2
(x)
p(s)W (s)

for s ≤ x ≤ x
2
,
(12.2.4.7)
where W (x)=y
1
(x)y

2
(x)–y

1
(x)y
2
(x) is the Wronskian determinant.
Remark. Formula (12.2.4.7) can also be used to construct the Green’s functions for the second, third, and
mixed boundary value problems. To this end, one should find two linearly independent solutions, y
1
= y
1
(x)
and y
2
= y
2
(x), of the homogeneous equation; the former satisfies the corresponding homogeneous boundary
condition at x = x
1
and the latter satisfies the one at x = x
2

.
12.2.5. Eigenvalue Problems
12.2.5-1. Sturm–Liouville problem.
Consider the second-order homogeneous linear differential equation
[p(x)y

x
]

x
+[λs(x)–q(x)]y = 0 (12.2.5.1)
subject to linear boundary conditions of the general form
α
1
y

x
+ β
1
y = 0 at x = x
1
,
α
2
y

x
+ β
2
y = 0 at x = x

2
.
(12.2.5.2)
It is assumed that the functions p, p

x
, s,and q are continuous, and p and s are positive
on an interval x
1
≤ x ≤ x
2
. It is also assumed that |α
1
| + |β
1
| > 0 and |α
2
| + |β
2
| > 0.
The Sturm–Liouville problem: Find the values λ
n
of the parameter λ at which problem
(12.2.5.1), (12.2.5.2) has a nontrivial solution. Such λ
n
are called eigenvalues and the cor-
responding solutions y
n
= y
n

(x) are called eigenfunctions of the Sturm–Liouville problem
(12.2.5.1), (12.2.5.2).
* The homogeneous boundary value problem—with r(x)=0 and a
1
= a
2
= 0—is assumed to have only
the trivial solution.
12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 483
12.2.5-2. General properties of the Sturm–Liouville problem (12.2.5.1), (12.2.5.2).
1
◦
.Thereareinfinitely (countably) many eigenvalues. All eigenvalues can be ordered so
that λ
1
< λ
2
< λ
3
< ···. Moreover, λ
n
→∞as n →∞; hence, there can only be a finite
number of negative eigenvalues. Each eigenvalue has multiplicity 1.
2
◦
. The eigenfunctions are defined up to a constant factor. Each eigenfunction y
n
(x)has
precisely n – 1 zeros on the open interval (x
1

, x
2
).
3
◦
. Any two eigenfunctions y
n
(x)andy
m
(x), n ≠ m, are orthogonal with weight s(x)
on the interval x
1
≤ x ≤ x
2
:

x
2
x
1
s(x)y
n
(x)y
m
(x) dx = 0 if n ≠ m.
4
◦
. An arbitrary function F (x) that has a continuous derivative and satisfies the boundary
conditions of the Sturm–Liouville problem can be decomposed into an absolutely and
uniformly convergent series in the eigenfunctions

F (x)=
∞

n=1
F
n
y
n
(x),
where the Fourier coefficients F
n
of F (x) are calculated by
F
n
=
1
y
n

2

x
2
x
1
s(x)F (x)y
n
(x) dx, y
n


2
=

x
2
x
1
s(x)y
2
n
(x) dx.
5
◦
. If the conditions
q(x) ≥ 0, α
1
β
1
≤ 0, α
2
β
2
≥ 0 (12.2.5.3)
hold true, there are no negative eigenvalues. If q ≡ 0 and β
1
= β
2
= 0, the least eigenvalue
is λ
1

= 0, to which there corresponds an eigenfunction y
1
= const. In the other cases where
conditions (12.2.5.3) are satisfied, all eigenvalues are positive.
6
◦
. The following asymptotic formula is valid for eigenvalues as n →∞:
λ
n
=
π
2
n
2
Δ
2
+ O(1), Δ =

x
2
x
1

s(x)
p(x)
dx.(12.2.5.4)
Paragraphs 12.2.5-3 through 12.2.5-6 will describe special properties of the Sturm–
Liouville problem that depend on the specific form of the boundary conditions.
Remark 1. Equation (12.2.5.1) can be reduced to the case where p(x) ≡ 1 and s(x) ≡ 1 by the change
of variables

ζ =


s(x)
p(x)
dx, u(ζ)=

p(x)s(x)

1/4
y(x).
In this case, the boundary conditions are transformed to boundary conditions of similar form.
Remark 2. The second-order linear equation
ϕ
2
(x)y

xx
+ ϕ
1
(x)y

x
+[λ + ϕ
0
(x)]y = 0
can be represented in the form of equation (12.2.5.1) where p(x), s(x), and q(x)aregivenby
p(x)=exp



ϕ
1
(x)
ϕ
2
(x)
dx

, s(x)=
1
ϕ
2
(x)
exp


ϕ
1
(x)
ϕ
2
(x)
dx

, q(x)=–
ϕ
0
(x)
ϕ
2

(x)
exp


ϕ
1
(x)
ϕ
2
(x)
dx

.
484 ORDINARY DIFFERENTIAL EQUATIONS
TABLE 12.2
Example estimates of the first eigenvalue λ
1
in Sturm–Liouville problems with boundary conditions of the first
kind y(0)=y(1)=0 obtained using the Rayleigh–Ritz principle [the right-hand side of relation (12.2.5.6)]
Equation
Test function
λ
1
, approximate
λ
1
, exact
y

xx

+ λ(1 + x
2
)
–2
y = 0
z =sinπx
15.337 15.0
y

xx
+ λ(4 – x
2
)
–2
y = 0
z =sinπx
135.317 134.837
[(1 + x)
–1
y

x
]

x
+ λy = 0
z =sinπx
7.003 6.772

√

1 + xy

x


x
+ λy = 0
z =sinπx
11.9956 11.8985
y

xx
+ λ(1 +sinπx)y = 0
z =sinπx
z = x(1 – x)
0.54105 π
2
0.55204 π
2
0.54032 π
2
0.54032 π
2
12.2.5-3. Problems with boundary conditions of the first kind.
Let us note some special properties of the Sturm–Liouville problem that is the first boundary
value problem for equation (12.2.5.1) with the boundary conditions
y = 0 at x = x
1
, y = 0 at x = x
2

.(12.2.5.5)
1
◦
.Forn →∞, the asymptotic relation (12.2.5.4) can be used to estimate the eigenval-
ues λ
n
. In this case, the asymptotic formula
y
n
(x)
y
n

=

4
Δ
2
p(x)s(x)

1/4
sin

πn
Δ

x
x
1


s(x)
p(x)
dx

+ O

1
n

, Δ =

x
2
x
1

s(x)
p(x)
dx
holds true for the eigenfunctions y
n
(x).
2
◦
.Ifq ≥ 0, the following upper estimate holds for the least eigenvalue (Rayleigh–Ritz
principle):
λ
1
≤


x
2
x
1

p(x)(z

x
)
2
+ q(x)z
2

dx

x
2
x
1
s(x)z
2
dx
,(12.2.5.6)
where z = z(x) is any twice differentiable function that satisfies the conditions z(x
1
)=
z(x
2
)=0. The equalityin (12.2.5.6) isattained if z =y
1

(x), where y
1
(x) is the eigenfunction
corresponding to the eigenvalue λ
1
. One can take z =(x–x
1
)(x
2
–x)orz =sin

π(x – x
1
)
x
2
– x
1

in (12.2.5.6) to obtain specific estimates.
It is significant to note that the left-hand side of (12.2.5.6) usually gives a fairly precise
estimate of the first eigenvalue (see Table 12.2).
3
◦
. The extension of the interval [x
1
, x
2
] leads to decreasing in eigenvalues.
4

◦
. Let the inequalities
0 < p
min
≤ p(x) ≤ p
max
, 0 < s
min
≤ s(x) ≤ s
max
, 0 < q
min
≤ q(x) ≤ q
max
be satisfied. Then the following bilateral estimates hold:
p
min
s
max
π
2
n
2
(x
2
– x
1
)
2
+

q
min
s
max
≤ λ
n
≤
p
max
s
min
π
2
n
2
(x
2
– x
1
)
2
+
q
max
s
min
.
12.2. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 485
5
◦

. In engineering calculations for eigenvalues, the approximate formula
λ
n
=
π
2
n
2
Δ
2
+
1
x
2
– x
1

x
2
x
1
q(x)
s(x)
dx, Δ =

x
2
x
1


s(x)
p(x)
dx (12.2.5.7)
may be quite useful. This formula provides an exact result if p(x)s(x) = const and
q(x)/s(x) = const (in particular, for constant equation coefficients, p = p
0
, q = q
0
,and
s = s
0
) and gives a correct asymptotic behavior of (12.2.5.4) for any p(x), q(x), and s(x).
In addition, relation (12.2.5.7) gives two correct leading asymptotic terms as n →∞if
p(x) = const and s(x)=const [andalsoif p(x)s(x) = const].
6
◦
. Suppose p(x)=s(x)=1 and the function q = q(x) has a continuous derivative.
The following asymptotic relations hold for eigenvalues λ
n
and eigenfunctions y
n
(x)as
n →∞:

λ
n
=
πn
x
2

– x
1
+
1
πn
Q(x
1
, x
2
)+O

1
n
2

,
y
n
(x)=sin
πn(x – x
1
)
x
2
– x
1
–
1
πn


(x
1
– x)Q(x, x
2
)+(x
2
– x)Q(x
1
, x)

cos
πn(x – x
1
)
x
2
– x
1
+ O

1
n
2

,
where
Q(u, v)=
1
2


v
u
q(x) dx.(12.2.5.8)
7
◦
. Let us consider the eigenvalue problem for the equation with a small parameter
y

xx
+[λ + εq(x)]y = 0 (ε → 0)
subject to the boundary conditions (12.2.5.5) with x
1
= 0 and x
2
= 1. We assume that
q(x)=q(–x).
This problem has the following eigenvalues and eigenfunctions:
λ
n
= π
2
n
2
– εA
nn
+
ε
2
π
2


k≠n
A
2
nk
n
2
– k
2
+ O(ε
3
), A
nk
= 2

1
0
q(x)sin(πnx)sin(πkx) dx;
y
n
(x)=
√
2 sin(πnx)–ε
√
2
π
2

k≠n
A

nk
n
2
– k
2
sin(πkx)+O(ε
2
).
Here, the summation is carried out over k from 1 to ∞. The next term in the expansion
of y
n
can be found in Nayfeh (1973).
12.2.5-4. Problems with boundary conditions of the second kind.
Let us note some special properties of the Sturm–Liouville problem that is the second
boundary value problem for equation (12.2.5.1) with the boundary conditions
y

x
= 0 at x = x
1
, y

x
= 0 at x = x
2
.
1
◦
.Ifq > 0, the upper estimate (12.2.5.6) is valid for the least eigenvalue, with z = z(x)
being any twice-differentiable function that satisfies the conditions z


x
(x
1
)=z

x
(x
2
)=0.
The equality in (12.2.5.6) is attained if z = y
1
(x), where y
1
(x) is the eigenfunction
corresponding to the eigenvalue λ
1
.