Differential Operators and the Elimination Method for Systems

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:

Expected Educational Results

Differential Operators

The Annihilator Method for Solving DEs

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

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

Investigation 12

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: .

Investigation 13

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

Investigation 14

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

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