Differential Operators and the Elimination Method for SystemsExpected Educational ResultsDifferential OperatorsSystems of Linear EquationsInvestigation 15Solving Systems of Linear EquationsInvestigation 16Investigation 17CC 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.

To convert -order ODE as a system of linear differential equations:

- Let , , \ldots, .
- Differentiate all . Note that .
- Convert all initial conditions, if any.
- Substitute all into the DE so that you have linear DEs in terms of .

**Example 05**:

Rewrite , , , into a system of linear equations.

**Solution**:

Let .

Then

and

by solving the third-order ode for .

For the initial conditions: , , and

The third-order ode can be written as a system of three linear odes:

Convert the following into a system of linear DEs.

- .
- .
- , , .

**Example 06**:

Solve the following system of linear odes:

**Solution**:

**Use Elimination to find **.

Apply to the first equation; and to the second equation:

**Subtract** the equations. Since the linear differential operators performed on are the same, is eliminated resulting in:

So,

,

[you can always find coefficients of the particular solution; here , ],

.

**Use Elimination to find **.

Apply to the first equation; and to the second equation:

**Subtract** the equations. Since the linear differential operators performed on are the same, is eliminated resulting in:

So,

[ is a different equation , so no need to consider linear independence between different solutions],

[you can always find coefficients of the particular solution; here , ], .

Thus, the solution to the system is:

Solve the following systems of linear DEs.

Solve the following systems of linear DEs.

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