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Differential operator

A differential operator considers differentiation as an abstract operation, which accepts a function and returns another.

 

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Explanation

We can write the differential operator with the letter D as

 ,   ,   ,  ... etc.

 

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Example 1

In the differential equation

we substitute the differential operator by D²

For (1 + D²)^–1 we use the binomium development, that gives

and because Dx⁵ = 5x⁴, D²x⁵ = 20x³, D³x⁵ = 60x², D⁴x⁵ = 120x, D⁵x⁵ = 120, D⁶x⁵ = 0, etc. we get

 

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Example 2

You can write the product rule with differential operators as

D(xψ) = (D(x))ψ + xD(ψ)

We convert this to

D(xψ) – xD(ψ) = (D(x))ψ

and since D(x) = 1 we get

D(xψ) – xD(ψ) = 1ψ

Operators can be treated as ordinary numbers

(Dx – xD)ψ = 1ψ

so that

Dx – xD = 1

 

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History The Britsch physicist Oliver Heaviside (1850 - 1925) has proposed this technique.

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