1. Class Notes 11: Power Series (3/3) Series Solution Singular Point
182A – Engineering Mathematics

2. Series Solution Ordinary & Singular Point
Ordinary Point (OP)
Singular Point (SP)
Regular Singular Point (RSP)
Irregular Singular Point (ISP)

3. Series Solution about Ordinary Point (OP)
Definition
Solution (Typically 2)
Example

4. Series Solution about Regular Singular Point (RSP)
Definition
Solution (Typically 2)
Example

5. RSP - Second solution Assume case 1, 2; case 3 Case 1 If are distinct
is positive non integer
Case 2
If are distinct
is a positive integer
constant that can be zero

6. RSP - Second solution (Continue)
Case 3

7. Series Solution about Non Regular Singular Point (NSP)
Definition

8. Classification of Singular Points – Example
which type of singular point
(Regular/Irregular)

9. Classification of Singular Points – Example
For x0=2
Factor (x-2)
Both functions are analytic at x=0
Regular Singular Point (RSP)
For x0=-2
Factor (x-(-2))
=x+2
Non-analytic at x0=-2
Irregular Singular Point (IRS)

10. Frobenius’ Theorem
If x= x0 is a regular singular point of the differential equation
Then there exists at least one solution of the form
where the number r is a constant to be determined.
The series will converge at least at some interval

11. Solution Protocol Substitute into the given differential equation
Determine the unknown coefficients cn using reduction relations
Find the unknown exponent r. (non-negative integer)
For the sake of simplicity in solving the differential equation, the regular singular point is x0=0
for any other regular singular point x0

12. RSP (Case 1) - Example (Two Series Solutions)
What type of point x0=0 is of the ideational equation?
Regular Point (RP)
Regular Singular Point (RSP)
Irregular Singular Point (IRS)
Conclusion - Regular Singular Point (RSP)

13. RSP (Case 1) - Example (Two Series Solutions)
Assuming a solution of the form
Plug the solution into the differential equation

14. RSP (Case 1) - Example (Two Series Solutions)

15. RSP (Case 1) - Example (Two Series Solutions)
Because nothing is gained by taking c0=0, we must then have
(*)
Plugging r1, r2 into (*)
(o)
(+)
Case 1 – Positive Non-Integer

16. RSP (Case 1) - Example (Two Series Solutions)
For (r1=2/3) Eq (o)
For (r2=0) Eq (+)
We encounter something that didn’t happen when we obtained solutions about an ordinary point. The two sets of coefficients contains the same multiple c0. We can omit the term c0

17. RSP (Case 1) - Example (Two Series Solutions)
Using the ratio test both y1(x) and y2(x) converge for all values of x
(except the origin)
y1(x) and y2(x) are linearly independent for all

18. RSP - Indicial Equation
Indicial roots or exponents
The indicial equation is typically quadratic equation
It is possible to obtain the indicial equation in advance of substituting
into the diff. eq.
of the singularity

19. RSP - Indicial Equation
Obtain the indicial equation in advance of substituting
into the diff. eq.
If x = 0 is a regular singular point of
Than
are analytic at x = 0
- That is the power series expansions
are valid or intervals that have a positive radius at convergence

20. RSP - Indicial Equation
- By multiplying by we get
After substituting and the two series for
we find that the general indicial equation to be
where are defined by

21. RSP (Case 2)- Example Indicial Equation (Direct Derivation)
The indicial equation is
Case 2 – Positive Integer

22. RSP (Case 2) - Example (only one series solution)

23. RSP (Case 2) - Example (only one series solution)

24. RSP (Case 2) - Example (only one series solution)

25. RSP (Case 2) - Second Solution
Finding a second solution ( positive integer case II)
We may or may not be able to find a second solution depends on
Indicial roots
Recurrence relation of
c may be equal to 0 if
We may use the following term for the second solution

26. RSP (Case 2) - Second Solution
Construct a second solution using the formula

27. Example (Finding a second solution)