1. Power Series Solutions of Linear DEs
Chapter 4
Power Series Solutions of Linear DEs

2. Learning Objective
At the end of the section, you should be able to solve DE with Power Series as solutions.

3. Power Series A power series in is an infinite series of the form
The above power series is centered at
x = a.

4. Power Series
center x = -1
center x = 0

5. Examples

6. Remark If the radius of convergence is R > 0, then is continuous
differentiable
integrable
over the interval (a-R, a+R).

7. Example
Given
Find

8. Example

9. Analytic at a Point A function is analytic at a point a
if it can be represented by a power series
in x-a:
with a positive or infinite radius
of convergence.

10. Adding two Power Series
Example
Write as a single summation.

11. Example
2 problems: exponents and starting indices

12. Example
Let

13. Example

14. Example

15. Example
Now same exponent
Yet to solve: first term!

16. Example

17. Exercise
Combine.

18. Solution 2 3 1
Let

19. Solution 1

20. Solution 2

21. Solution 3

22. Solution

23. Solution

24. Solution

25. Solution

26. Ordinary and Singular Point
A function is analytic at if
exists for any n

27. Ordinary and Singular Point
A point is said to be an ordinary
point of the DE if both and
are analytic at A point that is not an
ordinary point is said to be a singular
point of the DE.

28. Ordinary and Singular Point
Note:
If at least one of the function and
fails to be analytic at then
is a singular point.

29. Examples 1) Every finite value of is an ordinary point of the DE
2) is a singular point of the DE

30. Existence of Power Series Solutions
Theorem
If is an ordinary point of the DE,
we can always find two linearly independent
solutions in the form of a power series
centered at ( ).
Each series solution converges at least on some interval defined by where R is the distance from to the closest singular point

31. Example
Find a power series solution centered at 0 for the following DE

32. Example Ordinary points: All real numbers x.
Since there are no finite singular points,
The previous Theorem guarantees two power
series solutions centered at 0, and
convergent for

33. Let the solution be

37. Using the Identity Property:
for
The (recursive) relation generate consecutive
coefficients of the solution.

41. where

42. Example
Find a power series solution centered at 0 for the following DE

43. Example The standard form: Ordinary Points: All real numbers x.
Singular point: None.

44. Let the solution be

50. combine

57. Example
Find a power series solution centered at 0 for the following DE

58. Example

59. Example

60. Example

61. Example

62. Example
Using Identity Property:

63. Example

64. Example
Case 1:

65. Example
Case 2:

66. Example
From case 1:

67. Example
From case 2:

68. End