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SECTION 3.2 A BIT OF THEORY FOR SECOND ORDER LINEAR ODE’S
To answer the questions at the end of the last section we need a bit of theory, and we
may as well do this for the general second order homogeneous linear ODE
y
00
+
p
(
t
)
y
0
+
q
(
t
)
y
= 0
.
Notice that the coeﬃcient of
y
00
is 1. We abbreviate the lefthandside as
L
[
y
], and then we
call
L
a
diﬀerential operator.
If we let
D
stand for “take the derivative of” and then
D
2
stand for “take the second derivative of,” and so on, we can even write
L
=
D
2
+
p
(
t
)
D
+
q
(
t
),
though we must use this notation carefully because it mixes up algebraic multiplication and
application of the operator
D
.
EXISTENCE AND UNIQUENESS FACT.
This is carefully stated on page 144. The
heart of the matter is that we need continuity in an interval and initial values for both
y
and
y
0
at the same
t
0
in the interval. Then the solution of the initial value problem exists and is
unique and is deﬁned throughout the interval on which we have continuity. We do not need
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 Spring '08
 Fonken
 Differential Equations, Equations

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