Differential Operators and the Elimination Method for SystemsExpected Educational ResultsDifferential OperatorsThe Annihilator Method for Solving DEsInvestigation 12Investigation 13Investigation 14CC 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.

Steps for the Annihilator Method for Solving DEs. For example, solve

- Write in terms of .
- Find for
- Rewrite as . [Explain why ].
- Find the solutions to . [You will need to know which are coefficients of terms in . Why?]
- Write the general solution to the DE.

Use *linear differential operators* and *annihilators* to solve the following DEs:

**NOTE**: In each ode below, the **Method of Undetermined Coefficients** and **Variation of Parameters** may be easier to use. This question is to recognize that using linear differential operators and annihilators will result in the same solutions.

**Example 04**: Solve:

**Solution**:

Rewrite left-side of ode using *linear differential operator*, :

Find the *annihilator*, , for the nonhomogeneous function, , and apply to both sides of the ode:

The above equation is

Find the homogeneous solution :

The characteristic equation is and has roots . So, .

Find the particular solution :

From the annihilator, , . You **must** always use algebra to find the coefficients of , in this example .

Thus, the solution to the ode is: .

What do you notice about and the characteristic equation in **Example 04**? Explain.

What do you notice about and the particular solution? Explain.

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