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:

Linear differential equation

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

For Example-

Linear differential equation


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 –

  • Linear differential equation; The order of the highest order derivative in the given LDE is 2. So, it is a LDE of order 2.
  • Linear differential equation; 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 –

  • Linear differential equationIn 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:

Linear differential equation 0

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:

Linear differential equation in x

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 Linear differential equation 01 is:

Where, C = Arbitrary/Integration Constant

Here, the Integrating Factor, Integrating factor


The general solution of the linear differential equation of the standard form is:

Linear Differential equation in y

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:Linear differential equation

Solution:

LDE example

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


Example-2: Solve the LDE :

Solution:

LDE example 2

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


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