\[F\]
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.