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The D operator

Solving Differential Equations using the D operator
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Theory Of Differential Operator (differential Module)

Definition

A differential operator is an operator defined as a function of the differentiation operator.

It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another (in the style of a higher-order function in computer science).

The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include:

and if generalize

Note
is an operator and must therefore always be followed by some expression on which it operates.

Simple Equivalents

• means but
• Similarly and

The D Operator And The Fundamental Laws Of Algebra

The following differential equation:

may be expressed as: or

This can be factorised to give:
Examples

But is it justifiable to treat D in this way?

Algebraic procedures depend upon three laws.
• The Distributive Law:
• The Commutative Law:
• The Index Law:

If D satisfies these Laws, then it can be used as an Algebraic operator(or a linear operator). However:
• only when u is a constant.

Thus we can see that D does satisfy the Laws of Algebra very nearly except that it is not interchangeable with variables.

In the following analysis we will write

are constants and is a positive integer. As has been seen, we can factorise this or perform any operation depending upon the fundamental laws of Algebra.

We can now apply this principle to a number of applications.

The Use Of The D Operator To Find The Complementary Function For Linear Equations

It is required to solve the following equations:

Example:
Example - Simple example
Problem
Solve the following equation:-


Workings
Using the D operator this can be written as:-








Solution
Integrating using  as the factor



Three Useful Formulae Based On The Operator D

Equation A

Let represent a polynomial function

Since
and
From which it can be seen that:

Example:
Example - Equation A example
Problem

Workings
This can be re-written as:




Solution
We can put D = 4



Equation B

Where is any function of x

Applying Leibniz's theorem for the differential coefficient of a product.

Similarly and so on
therefore

Example:
Example - Equation B example
Problem
Find the Particular Integral of:

Workings



We have used D as if it were an algebraic constant but it is in fact an operator where 
Solution

Equation C - Trigonometrical Functions

And so on
Therefore

similarly

Example:
Example - Trigonometric example
Problem
Find the Particular Integral of:-

Workings
This can be re-written as:-


Using equation 1 we can put 



If we multiply the top and bottom of this equation by 



But 


Solution
But since 



Linear First Order D Equations With Constant Coefficients

These equations have on the right hand side

This equation is

Using an Integrating Factor of the equation becomes:-
Which is the General Solution.

Linear Second Order D Equations With Constant Coefficients

Where are the roots of the quadratic equation. i.e. the auxiliary equation.

Where is an arbitrary Constant

This equation can be re-written as:-

Integrating
• Thus when we can write the General Solution as:-

Where A and B are arbitrary Constants.
Example:
Example - Linear second order example
Problem





Workings


The roots of this equation are:-


Therefore the General Solution is


• The Special Case where 

From Equation (41)


or







• The roots of the Auxiliary Equation are complex.

If the roots of the are complex then the General Solution will be of the form , and the solution will be given by:-





Solution
The roots of this equation are :-



Physical Examples

Example:
Example - Small oscilations
Problem
Show that if  satisfies the differential equation  with k < n and if when 



The complete period of small oscillations of a simple pendulum is 2 secs. and the angular retardation due to air resistance is 0.04 X the angular velocity of the pendulum. The bob is held at rest so the the string makes a small angle  with the downwards vertical and then let go. Show that after 10 complete oscillations the string will make an angle of about 40' with the vertical.(LU)

Workings




Using the "D" operator we can write





When t = 0  = 0 and  = 0

and



Solution
At t = 0


We have been given that k = 0.02 and the time for ten oscillations is 20 secs.