Linear Differential Equation, Ordinary And Partial Equation, Important Topics For JEE 2024

Linear Differential Equation : Differential equation shows a relation between unknown functions and there derivative.it represents a set of particular family depends on the number of arbitrary constants in its Cartesian form.

Linear differential equation is defined as an equation having multiple of y or its derivative with respect to any independent variable as constant or functions of independent variable only such as is a linear differential equation because multiple of y is constant and multiple of is a function of independent variable, In general form this can be represented as

In above equation are the terms of independent variable and can be of any degree such as and while is a function of x and y ’, y ’’, y ’’’ are respectively the first, second, third derivative of dependent variable with respect to independent variable

Equation of the form will not be a linear differential equation if coefficient are function of dependent variable such as y or represents differential of dependent variable with respect to independent variable such as

Linear differential equation can be of two forms ordinary differential equation or partial differential equation

Ordinary Differential Equation

Ordinary Differential Equation: Ordinary differential follows the above-mentioned rule and contains derivative with respect to one independent variable only.

Partial Differential Equation

Partial Differential Equation: Partial differential follows the above-mentioned rule and contains derivative with respect to more than one independent variable such as . We have to discuss ordinary differential equation only. Linear differential equation which we have to discuss is of the form

Here denotes functions of independent variable x, let’s explore this differential equation in next section with the help of illustration and examples.

Linear Differential Equation Introduction

1. Find the integrating factor as I.F.=

2. Multiply it in whole equation as

3. Now integrate both sides with respect to x

Linear Differential Equation Example

Example 1: Solve the following differential equation

Sol. Given differential equation can be written as

Example 2: Solve the following differential equation

Sol. Given differential equation can be written as

Example 3: Solve the following differential equation given that when

Sol.

Rapid Question Based on linear differential equation

1. Solve the following differential equation given that when ?

2. Solve the following differential equation

Advance Illustrations based on linear differential equation

1. Solve the following differential equation ?

Sol.

2. Find the particular solution of the differential equation , given that ?

Sol. Compare the given differential equation with

1. Find the general solution of the differential equation and also find the equation of curve at the point (1, 1)?

2. Find the particular solution of the differential equation , given that ?