Notes On Basic Concepts of Differential Equations - CBSE Class 12 Maths

Differential equation:
An equation involving a derivative of the dependent variable with respect to the independent variable is called a differential equation.

Ex:

In this equation dv/dt = g - kv the variable, t is the independent variable and variable, v is the dependent variable.

dy/dx = y' , d2y/dx2 = y" , d3y/dx3 = y"' .............dny/dxn = yn


Classifying Differential Equations

Working order:

1. Find order

2. Find degree based on the order

We can determine the order and the degree of polynomial equations only.

The derivative with the highest order signifies the order of the differential equation.

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

The degree of the polynomial differential equation is the degree of the highest order derivative of the differential equation.

Ex:
Let (d3y/dx3)2 the order of the derivative, three, and the degree is two.

The equation is a third order, second degree equation.

Ex:

1) (d3y/dx3)2 + 2 (d2y/dx2)3 + 4 tan(dy/dx) = 0

This is not a polynomial differential equation. Hence, its degree cannot be determined.

2) x (dy/dx)2 + 6y = 0 First order second degree equation then the order of the equation is one and its degree is two.

The order and the degree (if defined) of a differential equation are always positive integers.

Summary

Differential equation:
An equation involving a derivative of the dependent variable with respect to the independent variable is called a differential equation.

Ex:

In this equation dv/dt = g - kv the variable, t is the independent variable and variable, v is the dependent variable.

dy/dx = y' , d2y/dx2 = y" , d3y/dx3 = y"' .............dny/dxn = yn


Classifying Differential Equations

Working order:

1. Find order

2. Find degree based on the order

We can determine the order and the degree of polynomial equations only.

The derivative with the highest order signifies the order of the differential equation.

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

The degree of the polynomial differential equation is the degree of the highest order derivative of the differential equation.

Ex:
Let (d3y/dx3)2 the order of the derivative, three, and the degree is two.

The equation is a third order, second degree equation.

Ex:

1) (d3y/dx3)2 + 2 (d2y/dx2)3 + 4 tan(dy/dx) = 0

This is not a polynomial differential equation. Hence, its degree cannot be determined.

2) x (dy/dx)2 + 6y = 0 First order second degree equation then the order of the equation is one and its degree is two.

The order and the degree (if defined) of a differential equation are always positive integers.

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