Remark
: The definition of a singular point assumes that the other coefficients do not
both vanish there, i.e., either
q
(
x
0
)
negationslash
= 0 or
r
(
x
0
)
negationslash
= 0.
If all three functions happen to
vanish at
x
0
, we can cancel any common factor (
x
−
x
0
)
k
, and hence, without loss of
generality, can assume at least one of the coefficients is nonzero at
x
0
.
Proofs of the basic existence theorem for differential equations at regular points can
be found in [
66
,
72
].
Theorem 12.5.
Let
x
0
be a regular point for the second order homogeneous linear
ordinary differential equation
(12.69)
. Then there exists a unique, analytic solution
u
(
x
)
to the initial value problem
u
(
x
0
) =
a,
u
′
(
x
0
) =
b.
(12
.
71)
The radius of convergence of the power series for
u
(
x
)
is at least as large as the distance
from the regular point
x
0
to the nearest singular point of the differential equation in the
complex plane.
Thus, every solution to an analytic differential equation at a regular point
x
0
can be
expanded in an convergent power series
u
(
x
) =
u
0
+
u
1
(
x
−
x
0
) +
u
2
(
x
−
x
0
)
2
+
· · ·
=
∞
summationdisplay
n
=0
u
n
(
x
−
x
0
)
n
.
(12
.
72)
Since the power series necessarily coincides with the Taylor series for
u
(
x
), its coefficients
†
u
n
=
u
(
n
)
(
x
0
)
n
!
are multiples of the derivatives of the function at the point
x
0
. In particular, the first two
coefficients
u
0
=
u
(
x
0
) =
a,
u
1
=
u
′
(
x
0
) =
b.
(12
.
73)
are fixed by the initial conditions.
Once the initial conditions have been specified, the
remaining coefficients must be uniquely prescribed thanks to the uniqueness of solutions
to initial value problems.
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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Vector Space, Berlin UBahn, 3 k

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