Differential Operators and the Elimination Method for SystemsExpected Educational ResultsDifferential OperatorsProperties of Linear Differential OperatorsInvestigation 10Investigation 11AnnihilatorsDefinition: Annihilator, Forms of AnnihilatorsCC BY-NC-SA 4.0

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

**Section 5.2**– Differential Operators and the Elimination Method for Systems

**Objective 15–1**: I can identify the linear differential operator for any polynomial function, exponential function, or sine and cosine functions.**Objective 15–2**: I can solve linear systems of ODEs using linear differential operators.

Let , , , be polynomial differential operators; and be sufficiently differentiable functions of ; and constants.

- Sum Rule:
- Product: .
- Linearity Rule:
- Multiplication Rule: If , then
- Commutativity of Multiplication: If and have constant coefficients, then .
- Substitution Rule:
- Exponential-Shift Rule:

Prove any property of linear differential operators.

- Show is equivalent to .
- Show is equivalent to .
- Show is
**not**equivalent to .

A linear differential operator, is said to **annihilate** a function if for all .

- For polynomials of degree ,
- For ,
- For ,
- For or ,
- For ,

**Note**: You do **not** to memorize all of the above . How can you succinctly remember the forms for each ?

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