The D operator
Solving Differential Equations using the D operator
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
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 higherorder 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.
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:
 only when u is a constant.
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
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 rewritten 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
y= \frac{1}{6}x^2 + \frac{5}{18}x + \frac{19}{108}}
Equation C  Trigonometrical Functions

And so on
Thereforesimilarly
Example:
Example  Trigonometric example
Problem
Find the Particular Integral of:
Workings
This can be rewritten 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 rewritten as: Integrating Thus when we can write the General Solution as:
Example:
Example  Linear second order example
Problem
Workings
The roots of this equation are:
Therefore the General Solution is
\;y + A\;e^{\frac{3 + \sqrt{5}}{2}x} + B\;e^{\frac{3\, \sqrt{5}}{2}x
 The Special Case where
 The roots of the Auxiliary Equation are complex.
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.
\therefore\;\;\;\;\theta _{10 cycles}\;=\;e^{0.02\times20}\left(cos\,\pi
\times20\;+\;\frac{0.02}{20}\;sin\,\pi \times20} \right)\frac{\pi }{180}\f]
\f[=\;e^{0.4}\times1\times\frac{\pi}{ 180}\;=\;\frac{1}{1.49182}\;\times\;\frac{\pi
}{180}times\;60\approx 40\,seconds