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1. GANDHINAGAR INSTITUTE OF
TECHONOLOGY(012)
SUBJECT : Advanced Engineering Mathematics
(2130002)
Active Learning Assignment on the topic of
“Power Series”
BE Mechanical
Prepared By: Yash Pandya
Guided By : Prof.

2. OVERVIEW
 Analytic function
 Ordinary points
 Singular points :
 Examples
 Regular singular points
 Irregular singular points

3. Analytic Function
 A function f(x) is said to be Analytic at x0 , if f(x) has Taylor series
expansion about x0 exists and converges to f(x) for all x in some open
interval including x0 .
 If a function f(x) is not analytic at x0 , then it is said to be Singular at
x0 .

4. Classification of singularities
2nd Order ODE
Ordinary Point Singular Point
Regular singular Point Irregular singular Point
If both
(x- x0)p(x) & q(x) Analytic at x0
If
p(x)&q(x) both Analytic at x0
If
p(x)&q(x) not Analytic at x0
If
(x- x0)p(x) & q(x) not Analytic at x0
 2
0xx   2
0xx 

5. Normalized form of function
0)()()( 2
2
 yxR
dx
dy
xQ
dx
yd
xP
 Where P(x), Q(x) & R(x) are polynomial in x.
 Assuming P(x) ≠ 0, then above equation can whiten in the normalized
form as…
0)()(2
2
 yxq
dx
dy
xp
dx
yd
 Where p(x) = Q(x)/P(x) and q(x) = R(x)/P(x)

6. Ordinary Point
 The point x0 is Ordinary point in the given equation…
0)()(2
2
 yxq
dx
dy
xp
dx
yd
 If p(x) = Q(x)/P(x) and q(x)= R(x)/P(x) both are analytic at x0.
 Then the given point x0 is called Ordinary Point.

7. Singular Point
 In the case of Ordinary points if p(x) or q(x) or both are not Analytic at x0.
 Then the point x0 is called Singular Point.
Regular Singular Point
 If both (x- x0)p(x) & q(x) are analytic at x0….
 Then singular point x0 of given equation is called Regular singular Point.
 2
0xx 
Irregular Singular Point
 If either (x- x0)p(x) or q(x) or both are not analytic at x0….
 Then singular point x0 of given equation is called Irregular singular Point.
 2
0xx 

8. Example
 Consider the equation:       0sincos2/
2
 yxyxyx 
 Here, x =  /2 is only singular point.
 So, x = x0 =  /2 .
 p(x) = & q(x) =
 We will demonstrate that x =  /2 is a regular singular point by showing that
the following functions are analytic at  /2.
 
 
 
 
x
x
x
x
x
x
x
x
x sin
2/
sin
2/,
2/
cos
2/
cos
2/ 2
2
2











 2
2/
cos
x
x
 2
2/
sin
x
x

9. Thank You