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Regular Singular Points
We now investigate the solution of the homogeneous secondorder linear equation
P(x)
d
2
y
dx
2
+
Q(x)
dy
dx
+
R(x)y
=
0
(1)
near a singular point. Recall that if the functions P(x), Q(x), and R(x) are polynomials
having no common factors, then the singular points of the equation (1) are simply those
points where P(x) = 0.
Example
:
1) The point x = 0 is the only singular point of the Bessel equation of order n,
x
2
y’’ + xy’ +(x
2
– n
2
)y = 0
2) The points x =1 and x = 1are the two singular points of the Legendre equation
(1 –x
2
)y’’ 2xy’ + n(n+1)y = 0
It turns out that some important features of the solutions of such equations are largely
determined by the behavior near their singular points.
Remark
: We will restrict our attention to the case in which x = 0 is a singular point of
the equation. A differential equation having x = a as a singular point is easily transform
by the substitution t = x – c into one having the corresponding singular point at 0.
Types of Singular Points
A differential equation having a singular point at 0, in general, will not have power series
solution of the form y(x) =
c
n
x
n
n
=
0
!
"
, so the straightforward method of the previous
section fails in this case.
To investigate the form that the solution of such equation might take, we assume that
equation (1) has analytic coefficient functions and we rewrite it in standard form,
d
2
y
dx
2
+
Q(x)
P(x)
dy
dx
+
R(x)
P(x)
y
=
0 or
d
2
y
dx
2
+
p(x)
dy
dx
+
q(x)y
=
0
where p(x)
=
Q(x)
P(x)
and q(x)
=
R(x)
P(x)
Recall that x = 0 is an ordinary point (as opposed to a singular point) of equation if both
p(x) and q(x) are analytic functions at x = 0, this means that they both have a power
series expansion around 0.
It can be proved that each of the functions p(x) and q(x) either is analytic or approach ±
∞
as x approaches 0. Consequently, x = 0 is a singular point of the equation provided that
either p(x) or q(x) (or both) approaches ±
∞
as x approaches 0.
Definition
: Given the normalized equation
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2
y
dx
2
+
Q(x)
P(x)
dy
dx
+
R(x)
P(x)
y
=
0 or
d
2
y
dx
2
+
p(x)
dy
dx
+
q(x)y
=
0
where p(x)
=
Q(x)
P(x)
and q(x)
=
R(x)
P(x)
let x
0
be a singular point.
If the functions
A(x) = (x – x
0
) p(x), and B(x) = (x – x
0
)
2
q(x)
are both analytic at x
0
, then x
0
is called a
regular singular point
of the equation.
If either (or both) of the functions A(x) or B(x) is not analytic at x
0
, then x
0
is called an
irregular singular
point.
Example
:
1) Given (1 + x)y’’ + 2xy’ – 3y = 0,
the normalized equation is
y''
+
2x
1
+
x
y'
!
3
1
+
x
y
=
0
Then, x = 1 is a singular point of the equation since p(x) and q(x) are not analytic at 1.
Since
A(x) = (x +1) p(x) = 2x, and B(x) = (x +1)
2
q(x) = 3(x + 1)
are both analytic functions at x = 1, then x = 1 is a regular singular point.
2) Given x
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 Fall '08
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