Differential Operators

Differential Operators
The Operator D:
The Operator is a transformation that transforms a function into another function. Hence differential
calculus involves an operator, the differential operator D, which transforms a (differentiable) function into
its derivative ,written as;
A solution of a differential equation is a function between the equation variables that satisfies the differential equation on some open interval; thus, y is a solution of eq.(1) if y is n times differentiable and real valued ;

? y?^((n))=f(x,y(x),y^ (x),y^? (x),…,y^(n-1) (x)) …………… (1)

for all x in some open interval (a, b) . In this case y is a solution of (1) on (a ,b).
Functions that satisfy a differential equation at isolated points are not interesting.
For example, y = x2 satisfies :

if and only if x = 0 or x = 1, but it’s not a solution of this differential equation because it does not
satisfy the equation on an open interval.
The graph of a solution of a differential equation is a solution curve. More generally, a curve C is said to be an integral curve of a differential equation if every function y= y(x) whose graph is a segment
of C is a solution of the differential equation. Thus, any solution curve of a differential equation is an
integral curve, but an integral curve need not be a solution curve.