3.2: FUNDAMENTAL SOLUTIONS OF LINEAR
HOMOGENEOUS EQUATIONS
KIAM HEONG KWA
We have learned in the last section that there are always two linearly
independent solutions
y
1
(
t
) and
y
2
(
t
) to a second-order homogeneous
linear equation
(1)
ay
00
+
by
0
+
cy
= 0
with constant coefficients.
Being linearly independent means neither
y
1
(
t
) nor
y
2
(
t
) is a constant multiple of the other. More formally,
y
1
(
t
)
and
y
2
(
t
) are said to be
linearly independent
on their common domain,
say
I
, if the linear relation
c
1
y
1
(
t
) +
c
2
y
2
(
t
) = 0, where
c
1
and
c
2
are
some constants, holds for all
t
∈
I
only if
c
1
=
c
2
= 0. Furthermore, in
this case, the general solution
y
(
t
) of (1) can be represented as a linear
combination of
y
1
(
t
) and
y
2
(
t
), i.e.,
y
(
t
) =
c
1
y
1
(
t
) +
c
2
y
2
(
t
), where
c
1
and
c
2
are integration constants.
More general statements are true. That is to say, there are always
two linearly independent solutions
y
1
(
t
) and
y
2
(
t
) to a general second-
order homogeneous linear equation
(2)
y
00
+
p
(
t
)
y
0
+
q
(
t
)
y
= 0
with continuous coefficients
p
(
t
) and
q
(
t
) on an open interval
I
.
In
addition, the general solution
y
(
t
) of (2) can be expressed as a
linear
combination
of
y
1
(
t
) and
y
2
(
t
), i.e.,
y
=
c
1
y
1
(
t
) +
c
2
y
2
, where
c
1
and
c
2
are integration constants. This raises the following questions.
(1) Why are there always two linearly independent solutions to (2)?
(2) Why the general solution of (2) can always be represented as a
linear combination of these linearly independent solutions?
The short answer to this questions is the fact that the
solution space
,
i.e., the collection of all solutions, of (2) forms a two-dimensional vector
space. The vectors are now twice-differentiable functions on the interval
I
. It is enlightening to compare these vectors to the vectors in the usual
two-dimensional Euclidean plane spanned by the basis vectors
i
and
j
.

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