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**:

**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.

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 .

A **linear operator of order ** is

for

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

**Example 01**:

**Example 02**:

**Example 03**:

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

- , ,

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

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

- , ,

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

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

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

, i.e., and

Explain.

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

Prove the following:

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

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