Differential Operators and the Elimination Method for Systems

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:

Expected Educational Results

Differential Operators

Systems of Linear Equations

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

  1. Let , , \ldots, .
  2. Differentiate all . Note that .
  3. Convert all initial conditions, if any.
  4. 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:

Investigation 15

Convert the following into a system of linear DEs.

  1. .
  2. .
  3. , , .

Solving Systems of Linear Equations

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:

Investigation 16

Solve the following systems of linear DEs.

Investigation 17

Solve the following systems of linear DEs.

CC BY-NC-SA 4.0

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