4initial value and boundary values problems

Some of the Ordinary Differential Equations :
The Ordinary differential equation has the form :
a_n (x) (d^n y)/(dx^(n ) )+a_(n-1) (x) (d^(n-1) y)/(dx^(n-1 ) )+?+a_0 (x)y=g(x) ………(1)

Where a_n (x) ,a_(n-1) (x) ,…,a_0 are coefficients, g(x) called the right hand side function . are all continuous real valued functions of x ,defined on an interval I and don t depend on y .
In eq.(1) if a_n (x)?0 in I then the eq. called a normal equation .
Otherwise it is called non-normal equation

Example : The eq.: (sin?(x)) (d^2 y)/?dx?^2 -3 (d^2 y)/?dx?^2 +e^x dy/dx+(cos?(x) )y=tan?(x)
Is normal on I=(0,?)
,be non-normal diff. eq. if I=[0,?] or I=[0,?)
If in eq.(1) g(x)=0 for all x?I is called homogeneous eq.
Otherwise is called nonhomogeneous eq.

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