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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.

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