Solving of Nonhomogeneous Linear Differential 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 ;
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;
For the higher derivatives:
And
And so on , For example; , , ….,etc
For a homogeneous linear ODE with constant coefficients, then
the second-order differential operator
I is the identity operator defined as Iy = y ,then
and for the n th-order written as;
Or can written as ;
Where
Which can formed by factorization for second order as ;
Where
Solving of Nonhomogeneous Linear Differential Equations ;
The nonhomogeneous diff. eq. with form second order ;
( )
( )
( ) ( ) ….(1)
Where ( ) and ( ) ( ) and ( ) are continuous functions for x on an interval I .
To find the general solution for nonhomogeneous eq.(1) ,it is necessary to;
1. Find the general solution of the associated homogeneous equation
( )
( )
( ) ….(2)
as a complementary function
2. Add to the particular solution of eq.(1) .
Thus there are several methods to find of eq.(1)
Variation of