## 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<br />
$L(Af+Bg)=AL(f)+BL(g)$
<br /> 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.<br /> 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
$\frac{d^2 f}{dx^2}+3 \frac{df}{dx}+2f=x$
.<br /> Expressing differentials as linear differential operators is often useful in transforming equations and using <a href="/university-maths-notes/elementary-calculus.html?amp;view=article&amp;id=1798&amp;catid=149">Laplace</a> and <a href="/university-maths-notes/elementary-calculus.html?amp;view=article&amp;id=1781&amp;catid=149">Fourier</a> Transforms.