Linear Differential Operators

A linear differential operator
\[F\]
is any differential operator that behaves as a linear operator. If
\[f, \: g\]
are functions and
\[A, \: B\]
are constants, then the operator
\[L\]
is linear if
\[L(Af+Bg)=AL(f)+BL(g)\]

The operator
\[\frac{d}{dx}\]
is linear and
\[\frac{d^n}{dx^n}\]
for any value of
\[n\]
.
\[\frac{\partial^2}{\partial x \partial y}\]
, and in fact partial derivatives of any order with respect to any variables are linear for continuously differentiable function.
Any differential equation of the form
\[L(f)=g\]
for some (usually unknown)
\[f\]
, and
\[g\]
is linear in
\[f\]
eg with
\[L=\frac{d^2}{dx^2}+3 \frac{d}{dx}+2\]
and
\[g(x)=x\]
  we have
\[\frac{d^2 f}{dx^2}+3 \frac{df}{dx}+2f=x\]
.
Expressing differentials as linear differential operators is often useful in transforming equations and using and Transforms.

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