Linear Differential Equations in n th-order ;

Linear Differential Equations in n th-order ;
An nth-order linear differential equation has the form ;
…. (1)
Where the coefficients ;(j=0,1,2,…,n-1) ,and depends solely on the variable (not on or any
derivative of )
If then Eq.(1) is homogenous ,if not then is nonhomogeneous .
…… 2
A linear differential equation has constant coefficients if all the coefficients are constants ,if one or
more is not constant then has variable coefficients .
Examples on linear differential equations ;
first order nonhomogeneous
second order homogeneous
third order nonhomogeneous
fifth order homogeneous
second order nonhomogeneous
Theorem 1: Consider the initial value problem given by the linear differential equation(1)and the n initial
Conditions;
… …(3)
Define the differential operator L(y) by
…. (4)
Where ;(j=0,1,2,…,n-1) is continuous on some interval of interest then
L(y) =
and a linear homogeneous differential equation written as;
L(y) =0
Definition: Linearly Independent Solution; A set of functions … is linearly
dependent on if there exists constants … not all zero ,such that
…. (5)
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