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.

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.