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4.1 Series Solutions
A power series is the sum of the infinite number of terms of the form

S = ao + a1(x – xo) + a2(x – xo) + … =
2
 a m (x  x o )m
m 0
where ao, a1, a2, … are constants, called the coefficients of the series. xo is a constant, called the
center of the series. A power series does not include terms with negative powers.
* The linear differential equations with constant coefficients always possess series
solutions.
The homogeneous solution of the linear differential equation with constant coefficients
y” + ay’ + by = r(x)
will have one of the type:
y = C 1 e  1 x + C2 e  2 x
y = (A + Bx) e x
y = (A cos x + B sin x) e x
All of these solutions can be expanded in power series of x

xm
x2
e =1+x+
+…= 
m!
2!
m 0
x
sin x = x -

( 1) m x 2 m 1
x3 x5
+
+…= 
( 2m  1)!
3!
5!
m 0
cos x = 1 -

( 1) m x 2 m
x2 x4
+
+…= 
( 2m)!
2!
4!
m 0
The series power form of y can be accepted as a solution provided that the differential equation
is satisfied by it and the series is convergent. Some differential equations with variable
coefficients possess series solutions.
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Some properties of power series
The n th partial sum of the series is
Sn(x) = ao + a1(x – xo) + a2(x – xo)2 + … + an(x – xo)n
and the remainder is
Rn(x) = an+1(x – xo)n+1 + an+2(x – xo)n+2 + …
A power series converges if Rn  0 as n  ; otherwise, it diverges
There is usually an interval over which the power series converges with the center at x =
xo; that is, the series converges if
|x – xo| < R
where R is called the radius of convergence. The radius of convergence can be obtained from
R = lim (m  )
EX:
am
a m 1
x m 1
x2
xm
e =1+x+
+…+
+
+…
( m  1)!
2!
m!
x
Radius of convergence
R
a
= lim (m  ) m = lim (m  )
a m 1
1
( m  1)!
m!
= lim (m  )
1
m!
( m  1)!
= lim (m  ) m  1 = 
A function y(x) is analytic at the point x = x o if it can be expressed as a power series

 a m ( x  x o ) m with R > 0.
m 0
If the functions p(x), q(x), and r(x) in the differential equation
y” + p(x)y’ + q(x)y = r(x)
are analytic at the point x = xo, the solution can be represented by a power series with a finite
radius of convergence, that is,
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
y(x) =
 a m ( x  x o ) m with R > 0
m 0
The point x = xo is called an ordinary (or regular) point.
EX:
y" = exy, every point x   is a regular point
x5y" = y, every point x except for x = 0 and x =  is a regular point
If p(x), q(x), or r(x) is not analytic at x = xo, the point x = xo is said to be a singular point.
Regular singular point and irregular singular point.
Consider a second order homogeneous linear equation
y” + p(x)y’ + q(x)y = 0
(4.1-1)
Regular singular point: The point x = xo is called a regular singular point of (4.1-1) if not both of
p(x), q(x) are analytic but both (x – xo)p(x) and (x – xo)2q(x) are analytic in the neighborhood of
xo.
Irregular singular point: The point x = xo is called an irregular singular point of (4.1-1) if it is
neither a regular point nor a regular singular point.
EX:
(a)
(x - 1)y" = y has a regular singular point at 1
(b)
x2y" + xy' = y has a regular singular point at 0
(c)
x3y" = (x + 1)y has an irregular singular point at 0
If x = xo is a regular point of the differential equation (4.1-1) then the power series method can
be applied. The general solution of Eq. (4.1-1) is y = Ay1(x) + By2(x) where y1 and y2 are

linearly independent series solutions (
 a m ( x  x o ) m ) which are analytic at x = xo. The radius
m 0
of convergence for each of the series solutions y1 and y2 is at least as large as the minimum of the
radii of convergence of the series for p(x) and q(x).
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