1
Lecture 14
We observe from the previous example how one can try to solve the initial
value problem
y
00
+
p
(
x
)
y
0
+
q
(
x
)
y
= 0;
y
(
x
0
) =
y
0
; y
0
(
x
0
) =
y
0
0
:
°
We need to know that the particular solution satisfying given initial
value problem is unique.
°
We should °nd two particular solutions
y
1
(
x
)
; y
2
(
x
)
satisfying the
ODE
y
00
+
p
(
x
)
y
0
+
q
(
x
)
y
= 0
:
At this point, we pay no attention
to initial conditions.
°
By principle of superposition,
y
(
x
) =
C
1
y
1
(
x
) +
C
2
y
2
(
x
)
is a solution.
Now we try to select constants
C
1
and
C
2
in such a way that the initial
conditions hold for
y
(
x
)
.
So, there are two important questions:
°
Is solution of the initial value problem unique?
°
What should we require from
y
1
and
y
2
to ensure that
C
1
and
C
2
can be
always selected so that the solution
y
(
x
) =
C
1
y
1
(
x
)+
C
2
y
2
(
x
)
satis°es
the initial condition?
Below, we will discuss the answers to these two fundamental questions.
1.1
Existence and uniqueness theorem for linear sec
ond order ODE
We give the following theorem without proof:
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 Fall '08
 Staff
 Differential Equations, Linear Algebra, Equations, Vector Space, Boundary value problem

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