Securing Higher Grades Costing Your Pocket? Book Your Assignment At The Lowest Price Now!

- +
250 Words
Upload File

    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, Dx, delx 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, Dt, delt.

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

    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 + a1yn-1 + a2yn-2 + ………..+an = q(x) where y is a function of x.

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

    dn/dxn y + a1 dn-1/dxn-1 y + ……. + an = q(x)

    or usinf D we get,

    (Dn + a1Dn-1 + ………….+ an) y = q(x)

    Now let us assume, Dn + a1 Dn-1 + ……. + an = 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.

    1. (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

    1. 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

    • Plagiarism Free Work

    • 24 X 7 Live Help

    • Services For All Subjects

    • Best Price Guarantee

    Live Reviews

    Ashley 02 Aug 2021

    I was not able to concentrate on academics and co-curricular activities all at once and hence I decided to seek professional assistance. The expert writers took care of my essay report so well as if it was their own. So no complaints at all!

    IOT 02 Aug 2021

    The service solution is extremely flexible that helps students to procure services anytime as they need one. I could connect to them at any time of the day which made me feel calm and relaxed.

    Ariella 02 Aug 2021

    My parents have been really happy with my academic performance lately, and all thanks to the team offering excellent support and academic help. Easy availability and a true choice of students around the world!

    Norah 02 Aug 2021

    I got a bonus mark this semester due to offering an error-free thesis. My examiner was so impressed with my assignment that she offered me a bonus mark which further boosted my grade.

    View All Reviews