3.Exact Equations

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.
Example
3.Exact Equations
A differential .equation :
M(x,y)dx+N(x,y)dy=0.
Is said to be exact if there exists a function f(x,y) such that
df(x,y)=M(x,y)dx+N(x,y)dy
Where M(x,y),N(x,y) and f(x,y) are continuous functions and have continuous first partial derivatives on some rectangle of the (x,y) plane .