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

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

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

Power Series center x = -1 center x = 0

Examples

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

Example Given Find

Example

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.

Adding two Power Series Example Write as a single summation.

Example 2 problems: exponents and starting indices

Example Let

Example

Example

Example Now same exponent Yet to solve: first term!

Example

Exercise Combine.

Solution 2 3 1 Let

Solution 1

Solution 2

Solution 3

Solution

Solution

Solution

Solution

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

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.

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

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

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

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

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

Let the solution be

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

where

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

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

Let the solution be

combine

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

Example

Example

Example

Example

Example Using Identity Property:

Example

Example Case 1:

Example Case 2:

Example From case 1:

Example From case 2:

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