Differential Operators and the Elimination Method for Systems

Differential Operators and the Elimination Method for SystemsExpected Educational ResultsDifferential OperatorsDefinition: Differential OperatorDefinition: Linear Differential Operators, Examples of Differential OperatorsIdentifying Linear Differential Operators, Investigation 01Investigation 02Investigation 03Investigation 04Investigation 05Investigation 06Investigation 07Investigation 08Investigation 09CC BY-NC-SA 4.0

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Differential Operators

Definition: Differential Operator

A differential operator is in the form, , where is the derivative for , i.e., .

NOTE: The differential operator always has a function as its argument. For example, is the representation of the second derivative of .

Definition: Linear Differential Operators,

A linear operator of order is

for

i.e., a Linear Differential Operator, is a linear combination of differential operators.

Examples of Differential Operators

Example 01:

Example 02:

Example 03:

Identifying Linear Differential Operators,

Investigation 01

Identify the lowest order for the following functions, so that . Explain.

  1. , ,

Investigation 02

Show the lowest order linear differential operator, , for an exponential function, , is , i.e., . Explain.

Investigation 03

Show the lowest order linear differential operator, , for the product of a polynomial function and exponential function, , is , i.e., . Explain.

  1. , ,

Investigation 04

Show the lowest order linear differential operator, , for either or , is , i.e., and . Explain.

Investigation 05

Show the lowest order linear differential operator, , for either or , is , i.e., and . Explain.

Investigation 06

Show the lowest order linear differential operator, , for either or , is

, i.e., and

Explain.

Investigation 07

Show the lowest order linear differential operator, , for either or , is , i.e., and . Explain.

Investigation 08

Prove the following:

Investigation 09

Using your work from above, identify for , , , and , , .

CC BY-NC-SA 4.0

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Last Modified: Monday, 19 October 2020 8:23 EDT