1 Differential operator

A differential operator $D$ takes some function as its input, performs operations (inv. differentiation), returning a new function. Notation: terms may not have dots, indicates product.

Any DE can be written in terms of a differential operator:

$L[X]=r(t)$ Take

D:=\frac{d^2(\cdot)}{dt}+t^2\frac{d(\cdot)}{dt}+\sin(t)( \cdot)$$ Then operator $D$ applied to $y$ returns $$D[y]=t^3$$ An example involving linear operator $D$ and function $x(t)$ $$D:= (\cdot)^2+t^2(\cdot)+\frac{d(\cdot)}{dt} \rightarrow D[x]=x^2+t^2x+x'$$