The D operator
Solving Differential Equations using the D operator
- 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
- 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
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:
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 EquationsIt is required to solve the following equations:
Example - Simple example
Solve the following equation:-
Using the D operator this can be written as:-
Integrating using as the factor
- Let represent a polynomial function
SinceandFrom which it can be seen that:
Example - Equation A example
This can be re-written as:
We can put D = 4
- Where is any function of x Applying Leibniz's theorem for the differential coefficient of a product.Similarly and so ontherefore
Example - Equation B example
Find the Particular Integral of:
We have used D as if it were an algebraic constant but it is in fact an operator where
Equation C - Trigonometrical Functions
- And so onThereforesimilarly
Example - Trigonometric example
Find the Particular Integral of:-
This can be re-written as:-
If we multiply the top and bottom of this equation by
Linear First Order D Equations With Constant CoefficientsThese 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:-
- Thus when we can write the General Solution as:-
Where A and B are arbitrary Constants.
Example - Linear second order example
The roots of this equation are:-
Therefore the General Solution is
- The Special Case where
- The roots of the Auxiliary Equation are complex.
The roots of this equation are :-
Example - Small oscilations
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)
Using the "D" operator we can write
When t = 0 = 0 and = 0
At t = 0
We have been given that k = 0.02 and the time for ten oscillations is 20 secs.