15
Homogeneous Linear Equations —
Verifying the Big Theorems
As promised, here we rigorously verify the claims made in the previous chapter. In a sense,
there are two parts to this chapter. The first is mainly concerned with proving the claims when
the differential equation in question is second order, and it occupies the first two sections. The
arguments in these sections are fairly elementary, though, perhaps, a bit lengthy. The rest of the
chapter deals with differential equations of arbitrary order, and uses more advanced ideas from
linear algebra.
If you’ve had an introductory course in linear algebra, just skip ahead to section 15.3 starting
on page 323. After all, the set of differential equations of arbitrary order includes the secondorder
equations.
If you’ve not had an introductory course in linear algebra, then you may have trouble fol
lowing some of the discussion in section 15.3. Concentrate, instead, on the development for
secondorder equations given in sections 15.1 and 15.2. You may even want to try to extend the
arguments given in those sections to deal with higherorder differential equations. It is “doable”,
but will probably take a good deal more space and work than we will spend in section 15.3 using
the more advanced notions from linear algebra.
And if you don’t care about ‘why’ the results in the previous chapter are true, and are blindly
willing to accept the claims made there, then you can skip this chapter entirely.
15.1
FirstOrder Equations
While our main interest is with higherorder homogeneous differential equations, it is worth
spending a little time looking at the general solutions to the corresponding firstorder equations.
After all, via reduction of order, we can reduce the solving of secondorder linear equations to
that of solving firstorder linear equations. Naturally, we will confirm that our general suspicions
hold at least for firstorder equations. More importantly, though, we will discover a property of
these solutions that, perhaps surprisingly, will play a major role is discussing linear independence
for sets of solutions to higherorder differential equations.
With
N
=
1 the generic equation describing any
N
th
order homogeneous linear differential
equation reduces to
A
dy
dx
+
By
=
0
311
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312
Homogeneous Linear Equations — Verifying the Big Theorems
where
A
and
B
are functions of
x
on some open interval of interest
I
(using
A
and
B
instead
of
a
0
and
a
1
will prevent confusion later). We will assume
A
and
B
are continuous functions
on
I
, and that
A
is never zero on that interval. Since the order is one, we suspect that the
general solution (on
I
) is given by
y
(
x
)
=
c
1
y
1
(
x
)
where
y
1
is any one particular solution and
c
1
is an arbitrary constant. This, in turn, corresponds
to a fundamental set of solutions — a linearly independent set of particular solutions whose linear
combinations generate all other solutions — being just the singleton set
{
y
1
}
.
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 Summer '09
 Differential Equations, Linear Equations, Equations, Derivative, Vector Space, yk, Wronskians, N solutions

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