The D operator - Differential

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Solving Differential Equations using the D operator

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Contents

Theory Of Differential Operator (differential Module)
The D Operator And The Fundamental Laws Of Algebra
The Use Of The D Operator To Find The Complementary Function For Linear Equations
Three Useful Formulae Based On The Operator D
Linear First Order D Equations With Constant Coefficients
Linear Second Order D Equations With Constant Coefficients
Physical Examples
Page Comments

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:

The D operator - Differential - Calculus (2)

and if generalize

The D operator - Differential - Calculus (3)

Note

The D operator - Differential - Calculus (4)

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

The D operator - Differential - Calculus (40)

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

The D operator - Differential - Calculus (52)

is any function of xApplying Leibniz's theorem for the

The D operator - Differential - Calculus (53)

differential coefficient of a product.

Similarly

The D operator - Differential - Calculus (58)

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

Example - Trigonometric example

Problem

Find the Particular Integral of:-

Workings

This can be re-written as:-

(1)

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)

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 angularretardation due to air resistance is 0.04 X the angular velocity of the pendulum. The bob is heldat rest so the the string makes a small angle with the downwards vertical andthen let go. Show that after 10 complete oscillations the string will make an angle of about 40'with the vertical.(LU)