Series Solutions Near an Ordinary Point
We are considering methods of solving second order linear equations when the
coefficients are functions of the independent variable.
We consider the second order linear homogeneous equation
P(x)
d
2
y
dx
2
+
Q(x)
dy
dx
+
R(x)y
=
0
(1)
since the procedure for the nonhomogeneous equation is similar.
Many problems in mathematical physics lead to equations of this form having polynomial
coefficients; examples include the Bessel equation
x
2
y’’ + xy’ + (x
2
– a
2
) y = 0
where a is a constant, and the Legendre equation
(1 – x
2
) y’’ – 2xy’ + c(c + 1) y = 0
where c is a constant.
For this reason, as well as to simplify the algebraic computations, we primarily consider
the case in which P, Q, and R are polynomials. However, we will see that the method can
be applied when P, Q, and R are analytic functions.
For the time being, we assume P, Q, and R are polynomials and they do not have
common factors. Suppose that we wish to solve equation (1) in a neighborhood of a point
a. The solution of equation (1) in an interval containing point a is closely related with the
behavior of P(x) in the interval.
Definition
: Given the equation
P(x)
d
2
y
dx
2
+
Q(x)
dy
dx
+
R(x)y
=
0
the equation
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)
is called the
equivalent normalized form
of equation(1).
Definition
: The point a is called an
ordinary point
of equation (1) if both of the
functions p(x) and q(x) in the equivalent normalized form, are analytic functions at the
point a.
If either or both of these functions are not analytic at a, then the point a is a
singular
point
of equation (1).
Examples
:
1) Given y’’ + xy’ + (x
2
+ 2) = 0
since p(x) = x and q(x) = x
2
+ 2 are polynomials, then they are analytic functions
everywhere, then every real number is an ordinary point.
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View Full Document2) Given
(x
2
!
1)y''
+
xy'
+
1
x
y
=
0
(x
2
– 1)
the equivalent normalized equation is
y''
+
x
x
2
!
1
y'
+
1
x x
2
!
1
( )
y
=
0
where p(x)
=
x
x
2
!
1
and q(x)
=
1
x x
2
!
1
( )
then the points 1, 1, and 0 are singular points of the equation, any other real number is an
ordinary point of the equation.
Theorem
: If a is an ordinary point of the differential equation (1),
then p(x) = Q(x)/P(x) and q(x) = R(x)/P(x) are analytic at a, and there are two nontrivial
linearly independent power series solutions of equation (1) of the form
c
n
(x
!
a)
n
,
n
=
0
"
#
d
n
(x
!
a)
n
n
=
0
"
#
Furthermore, the radius of convergence for each of the series solutions is at least as large
as the minimum of the radii of convergence of the series for p(x) and q(x).
Example
:
1) The point a = 0 is an ordinary point of the equation
y’’ + (x
3
+ 1) y’ – xy = 0
then, the differential equation has a solution in the form
c
n
x
n
n
=
0
!
"
whose radius of
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 Fall '08
 STAFF
 Trigraph, CN

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