### 8.2.2. 2^{nd} Order LDE

LDEs of the 2^{nd} order are of the
form

y
′
′
+
P
1
(
x
)
y
′
+
P
2
(
x
)y=R(
x
)

If the coefficients *P*_{1}, *P*_{2}
and *R* are continuous over some open
interval *J*, then an existence theorem
guarantees that solutions always exists over *J* [see §8.5]. However, there
is no general formula analogous to (8.2) so that the theory is far from
complete, except in special cases.

#### Theorem 8.2.

Consider the DE

y
′
′
+a
y
′
+by=0
(8.3)

where *a*
and *b* are real constants. Every solution of (8.3) on
(
−∞,∞
)
has the form

y=
e
−ax/2
[
c
1
u
1
(
x
)+
c
2
u
2
(
x
)
]
(8.4)

where *c*_{1}
and *c*_{2} are constants and
the functions *u*_{1} and *u*_{2} are determined according
to the sign of the **discriminant**
d=
a
2
−4b
as follows:

(a) If
d=0
,
then
u
1
(
x
)=1
and
u
2
(
x
)=x
.

(b) If
d>0
,
then
u
1
(
x
)=
e
kx
and
u
2
(
x
)=
e
−kx
,
where
k=
1
2
d
.

(c) If
d<0
,
then
u
1
(
x
)=coskx
and
u
2
(
x
)=sinkx
,
where
k=
1
2
−d
.

#### Comment

By replacing each *n*th order derivative *y*^{(n)} by the *n*th power monomial *r*^{n},
we turn a LDE with constant coefficients into a polynomial equation, known as
the **characteristic equation** of the
LDE. For eq(8.3), this is

r
2
+ar+b=0
(8.5)

with roots

r
1
=
1
2
(
−a+
d
)
and
r
2
=
1
2
(
−a−
d
)

Thus, *d*
is so-called because it is actually the discriminant of (8.5) that determines
the nature of its solutions.

Assuming we are working with complex
numbers, it is then easily shown that all solutions of a LDE with constant
coefficients are linear combinations of functions of the form
f(
x
)=
e
r x
,
where *r* is a root of the
characteristic equation of the LDE.
Theorem 8.2 then follows by re-expressing everything in terms of real
functions using

1
2
(
e
ikx
±
e
−ikx
)={
coskx
isinkx