Table of Content

Differential operator

Differential operator takes an action of finding derivative of a function.If we have y = f(x) then the derivative of y with respect to xis represented as, y’ = f’(x)

Or d/dx y = d/dx f(x) here d/dx is a differential operator and it takes an action of getting derivative of f(x) with respect to x.

Various forms to represent a differential operator:

d/dx , D, D_x, del_x all these operators are used for first order derivatives of a function with respect to ‘x’.

in case of different variable i.e. t we can represent these operators as, d/dt, D, D_t, del_t.

for nth order derivatives we mathematicsrepresent the differential operator using such notations: dⁿ/dxⁿ, Dⁿ, D_xⁿ

Derivative of a function f(x) can be written as: d/dx f(x) = f ‘(x) or [f(x)]’.

Linear differential operators with constant coefficient:

Let us consider an ordinary differential equation:

yⁿ + a₁y^n-1 + a₂y^n-2 + ………..+a_n = q(x) where y is a function of x.

using differential operator d/dx we can write this equation as:

dⁿ/dxⁿ y + a₁ d^n-1/dx^n-1 y + ……. + a_n = q(x)

or usinf D we get,

(Dⁿ + a₁D^n-1 + ………….+ a_n) y = q(x)

Now let us assume, Dⁿ + a₁ D^n-1 + ……. + a_n = p(D)

Where, p(D) represents a polynomial of differential operators with constant coefficient.

Hence we get more simple form as: p(D) y = q(x)

Here p(D) acts as a operator and operates on a function y = f(x).

Rules of operators

Sum rule: differentiation is linear

D(f + g) = D(f) + D(g)

D(f – g) = D(f) – D(g)

Constant factor rule:

D(k * f) = k* D(f)

Composition rule:

(D₁ o D₂)(f) = D₁(D₂(f))

xD is not equal to Dx.

Linearity rule:

D(c₁f + c₂g) = c₁D(f) + c₂D(g)

Solved examples using polynomial differential operator.

(D² – 5D + 6)y = 0

Characteristic equation for the above ordinary differential equation is: r² – 5r + 6 = 0

Factorizing characteristic function we get,

(r – 3)(r – 2) = 0

Gives, r = 3 and r = 2

Hence general solution we get as: c₁ * e^3x + c₂ * e^2x

Non-homogenous equation:

(D² – 3D + 2)y = sin 2x

Characteristic equation: r² – 3r + 2 = 0

Factorizing this equation we get,

(r – 2)(r – 1) = 0

Gives, r = 2 and r =1

Hence general solution we get as: c₁ * e^x + c₂ * e^2x

Particular solution for q(t) = sin t is of the form: A sin t + B cos t

Hence particular solution: c₃ sin 2x + c₄ cos 2x

Solution for the above give ordinary differential equation:

c₁ * e^x + c₂ * e^2x + c₃ * sin 2x + c₄ * cos 2x

Differential operator ‘del’ is an important vector differential operator.Del (triangular) = x del/ del x + y del/ del y + z del/ delz. Operator del is used to find divergence, curl and gradient of various objects.

Our Amazing Features

On Time Delivery
There is no deadline that can stop our writers from delivering quality assignments to the students.
Plagiarism Free Work
Get authentic and unique assignments by using our 100% plagiarism-free services.
24 X 7 Live Help
The experienced team of Live web tutors has got your back at all times of the day. Get connected with our customer support executives to receive instant and live solutions for your assignment problems.
Services For All Subjects
We can build quality assignments in the subjects you're passionate about. Be it Engineering and Literature or Law and Marketing we have an expert writer for all.
Best Price Guarantee
Get premium service at a pocket-friendly rate. At live web tutors, we understand the tight budget of students and thus offer our services at highly affordable prices.

FREE RESOURCES

FREE SAMPLE FILE

live review Our Mission Client Satisfaction

Avoid failure and the consequences; send the thesis to them anytime, as they offer expert editing services.

18 Jan 2021

Harmoni
With the best values and adherence, the team upholds their promises and does everything possible to offer you the best. You can reach out without a second thought.

18 Jan 2021

Heather
There are distinctive teams specializing in distinctive forte, while some in writing, some in editing. Hence, you have got the best of the deals in one single place.

18 Jan 2021

Saoirse
The information that you provide are adhered by the strictest confidence and privacy. So thus they never give away the details or reveal anything you offer.

18 Jan 2021

Azaria
The writers have a strong competence in the language and offer writing solutions accordingly. Hence, prepare yourself for attaining the most exceptional paper in hand.

18 Jan 2021

Alisha

View All Review