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.