Chapter 4
Series Solution of Di!erential
Equations and Special Functions
4.1 Introduction
Many physical problems give rise to di!erential equations that are linear but have variable
coe”cients. Unfortunately, such equations do not permit a general solution in terms of
known functions. Although numerical methods can be used to solve them, it is often more
convenient to find a solution in the form of an infinite convergent series.
The series solution of certain di!erential equations leads to special functions such as
Bessel’s function, Legendre’s polynomial, Laguerre’s polynomial, Hermite’s
polynomial, and Chebyshev polynomials. These special functions have numerous
engineering applications.
Consider the following second-order ordinary di!erential equation with variable coef-
ficients.
d2 y dy
P0 (x) + P 1 (x) + P2 (x) y = 0 (4.1)
dx2 dx
where P0 (x),P1 (x), and P2 (x) are polynomials of x.
Definition 4.1. 1. If P0 (a) →= 0 then x = a is called an ordinary point of (4.1),
otherwise a is a singular point.
2. A singular point x = a of (4.1) is called a regular singular point, if (4.1) is put
in the form
d2 y Q1 (x) dy Q2 (x)
+ + y=0
dx 2 (x ↑ a) dx (x ↑ a)2
where Q1 (x) and Q2 (x) process derivatives of all orders in a neighborhood of a.
34
, 3. A singular point that is not regular is called an irregular singular point.
Example 4.2. Identify singular points of
x2 (x ↑ 2)2 y →→ + 2(x ↑ 2)y → + (x + 3)y = 0.
Determine whether they are regular or irregular singular points.
4.2 Series Solutions
The general solution of a linear di!erential equation of second order will consist of two
series, say y1 and y2 . Then the general solution will be y = ay1 + by2 where a and b are
arbitrary constants.
4.2.1 Solution about Ordinary Points
The solution for the equation (4.1) when x = 0 can be determined as follows.
Step 1: Assume its solution to be of the form
↑
!
y = a0 + a1 x + a2 x2 + · · · + an xn + · · · = an xn (4.2)
n=0
dy d2 y dy d2 y
Step 2: Calculate and from (4.2) and substitute the values of y, , and
dx dx2 dx dx2
in (4.1).
Step 3: Equate to zero the coe”cients of the various powers of x and determine a2 , a3 ,
a4 · · · in terms of a0 and a1 . The result obtained by equating the coe”cient of xn to zero
is called the recurrence relation.
Step 4: Substituting the values of a2 , a3 , a4 · · · in (4.2), we get the desired series solution
having a0 and a1 as its arbitrary constants.
Example 4.3. Solve the following ODEs assuming they have a series solution.
d2 y
1. +x y =0
dx2
2. y →→ + xy → + y = 0
Exercise 4.4. Solve the following ODEs.
d2 y
1. ↑y =0
dx2
d2 y
2. + x2 y = 0
dx2
35
, Example 4 2 .
22(x -
2) 2y" + 2(x -
2)y+ ( 3)y + = 0
Po(x) =
x2(x -
2)2
=>
Singular point x = 0
,x = 2
y" + 2(x -
2)y + (x + 3)y = 0
x(x-2)2 43(2-2)2
y ( + b)
+y +
x2(x-2
=
Consider n = 0
;
Q , (x) = 2
and Q2(x) = (+ 3)
x(x 2)
(x 2)
-
-
=> & , (x) is not continuous at x = 0 -
=> P , (x) is not differentiable at x = 0 .
=> x = 0 is an
irregular singular point .
Consider x = 2 ;
Qi(x) = 2
& (x) =
n+ 3
x2 x2
=> Continous at x=2
=> Differentiable at x= 2
=>
regular singular point
x= 2 is a
Series Solution of Di!erential
Equations and Special Functions
4.1 Introduction
Many physical problems give rise to di!erential equations that are linear but have variable
coe”cients. Unfortunately, such equations do not permit a general solution in terms of
known functions. Although numerical methods can be used to solve them, it is often more
convenient to find a solution in the form of an infinite convergent series.
The series solution of certain di!erential equations leads to special functions such as
Bessel’s function, Legendre’s polynomial, Laguerre’s polynomial, Hermite’s
polynomial, and Chebyshev polynomials. These special functions have numerous
engineering applications.
Consider the following second-order ordinary di!erential equation with variable coef-
ficients.
d2 y dy
P0 (x) + P 1 (x) + P2 (x) y = 0 (4.1)
dx2 dx
where P0 (x),P1 (x), and P2 (x) are polynomials of x.
Definition 4.1. 1. If P0 (a) →= 0 then x = a is called an ordinary point of (4.1),
otherwise a is a singular point.
2. A singular point x = a of (4.1) is called a regular singular point, if (4.1) is put
in the form
d2 y Q1 (x) dy Q2 (x)
+ + y=0
dx 2 (x ↑ a) dx (x ↑ a)2
where Q1 (x) and Q2 (x) process derivatives of all orders in a neighborhood of a.
34
, 3. A singular point that is not regular is called an irregular singular point.
Example 4.2. Identify singular points of
x2 (x ↑ 2)2 y →→ + 2(x ↑ 2)y → + (x + 3)y = 0.
Determine whether they are regular or irregular singular points.
4.2 Series Solutions
The general solution of a linear di!erential equation of second order will consist of two
series, say y1 and y2 . Then the general solution will be y = ay1 + by2 where a and b are
arbitrary constants.
4.2.1 Solution about Ordinary Points
The solution for the equation (4.1) when x = 0 can be determined as follows.
Step 1: Assume its solution to be of the form
↑
!
y = a0 + a1 x + a2 x2 + · · · + an xn + · · · = an xn (4.2)
n=0
dy d2 y dy d2 y
Step 2: Calculate and from (4.2) and substitute the values of y, , and
dx dx2 dx dx2
in (4.1).
Step 3: Equate to zero the coe”cients of the various powers of x and determine a2 , a3 ,
a4 · · · in terms of a0 and a1 . The result obtained by equating the coe”cient of xn to zero
is called the recurrence relation.
Step 4: Substituting the values of a2 , a3 , a4 · · · in (4.2), we get the desired series solution
having a0 and a1 as its arbitrary constants.
Example 4.3. Solve the following ODEs assuming they have a series solution.
d2 y
1. +x y =0
dx2
2. y →→ + xy → + y = 0
Exercise 4.4. Solve the following ODEs.
d2 y
1. ↑y =0
dx2
d2 y
2. + x2 y = 0
dx2
35
, Example 4 2 .
22(x -
2) 2y" + 2(x -
2)y+ ( 3)y + = 0
Po(x) =
x2(x -
2)2
=>
Singular point x = 0
,x = 2
y" + 2(x -
2)y + (x + 3)y = 0
x(x-2)2 43(2-2)2
y ( + b)
+y +
x2(x-2
=
Consider n = 0
;
Q , (x) = 2
and Q2(x) = (+ 3)
x(x 2)
(x 2)
-
-
=> & , (x) is not continuous at x = 0 -
=> P , (x) is not differentiable at x = 0 .
=> x = 0 is an
irregular singular point .
Consider x = 2 ;
Qi(x) = 2
& (x) =
n+ 3
x2 x2
=> Continous at x=2
=> Differentiable at x= 2
=>
regular singular point
x= 2 is a
