Method of Solving Homogeneous Equations with Constant Coefficients

Method of Solving Homogeneous Equations with Constant Coefficients

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 ;
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 ;


Which can formed by factorization for second order as ;