A **differential equation** is an equation containing an independent variable, dependent variable and differential coefficients of dependent variable with respect to independent variable, ** Linear differential equations** is a type of differential equation on which we will focus on this blog.

Author – Ojasvi Chaplot

## Introduction to Linear Differential Equation(LDE)

A ** linear differential equation** is a differential equation, having degree 1, that consists of an independent variable, dependent variable and differential coefficients of dependent variable with respect to independent variable.

Furthermore, a linear equation which consists of one or more terms having the derivatives of the dependent variable with respect to one or more independent variables is called as a *linear differential equation** (LDE)*.

## Standard form and Representation of Linear Differential Equation

The standard form and the representation of a LDE is:

Where, P and Q are either numeric constants or functions of x in the LDE.

For Example-

## General Terminologies

### Order of a Differential Equation

The order of a differential equation is the order of the highest order derivative appearing in the equation.

For Example –

; The order of the highest order derivative in the given LDE is 2. So, it is a LDE of order 2. ; Similarly, in this equation, the order of the highest order derivative is 1. So, it is a LDE of order 1.

** GENERAL NOTE:** Remember that the order of a LDE (or any differential equation) is a positive integer.

### Degree of Differential Equation

The degree of a differential equation is the degree of the highest order derivative when differential coefficients are made from radicals and fractions.

The degree of a LDE is always 1; otherwise, it is a non-linear differential equation.

For Example –

In this equation, the degree of the highest order derivative is 1. So, it is a LDE of degree 1.

### Non-Linear Differential Equation

A differential equation is said to be a non-LDE, if:

- The degree of the differential equation is more than 1.
- Any of the differential coefficient has exponent more than 1.
- The exponent of the dependent variable is more than 1.
- The terms containing the products of dependent variable and its differential coefficients are present in the differential equation.

#### Generally, the standard form of a LDE in y is:

Where, P and Q are either numeric constants or functions of x in the LDE..

#### And, the standard form of a LDE in x is:

Where, P and Q are either numeric constants or functions of y in the LDE.

## Formula for the General Solution of the Linear Differential Equation

The important formulas to find out the general solution of the LDE(s) are listed below.

The general solution of the *linear differential equation* of the standard form

Where, C = Arbitrary/Integration Constant

Here, the Integrating Factor,

The general solution of the *linear differential equation* of the standard form

Where, C = Arbitrary/Integration Constant

Here, the Integrating Factor,

## Steps to Find the Solution of a LDE

The steps involved in finding out the solution of the LDE(s) are given as in the following sequence:

- First, rearrange all the terms of the given equation in the standard form of LDE; where P and Q are constants or functions of the independent variable only.
- The step that comes next is to obtain the Integrating Factor (I.F) by using the formula of Integrating Factor (I.F) corresponding to the respective standard form.
- Now, put the value of Integrating Factor (I.F) in the formula for the general solution of a LDE and then solve the obtained equation and simplify it.

Therefore, the solution of the given LDE is obtained.

## Illustrative Examples-

** Example-1: **Find the solution of the linear differential equation:

*Solution:*

** Answer:** y =x³+Cx ; where C = Arbitrary/Integration Constant.

** Example-2: **Solve the LDE :

*Solution:*

Therefore, ** Answer:** y =2x²+Cx ; where C = Arbitrary/Integration Constant.

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