## 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&id=1798&catid=149">Laplace</a> and <a href="/university-maths-notes/elementary-calculus.html?amp;view=article&id=1781&catid=149">Fourier</a> Transforms.