## Ordinary Differential Equation

__Ordinary Differential Equation:__

Higher order differential equations with constant coefficients – Method of variation of parameters – Cauchy’s and Legendre’s linear equations – Simultaneous first order linear equations with constant coefficients.

The study of a differential equation in applied mathematics consists of three phases.

(**i**) Formation of differential equation from the given physical situation, called modeling.

(**ii**) Solutions of this differential equation, evaluating the arbitrary constants from the given conditions, and

(**iii**) Physical interpretation of the solution.

__Higher Order Linear Differential Equation With Constant Coefficient:__

General form of Linear differential equation of nth order with constant coefficients is:

Where X is a function of x and a_{1},a_{2.}...,a_{n} are constant and is called Linear Differential Equations of nth order with constant coefficients. Since the highest order of the derivative appearing in (1) is n, it is called a differential equation of n^{th} order and it is called linear. Using the familiar notation of differential operators:

Then (1) can be written in the form

Here f (D) is a polynomial of degree n in D

If x = 0, the equation

f (D) y = 0

is called a homogeneous equation. If x ≠ 0 then the Eqn. (2) is called a non-homogeneous equation.

**(1) The general form of the differential equation of second order is:
**

Where P and Q are constant and R is a function of X or constant.

**(2) Differential Operators:
**

The symbol D stands for the operation of differential