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Description of Order and Degree Of Differential Equation

How To Determine The Order Of Differential Equation And Its Degree?

Order of Differential Equation

The number of the highest derivative in a differential equation.

A differential equation of order 1 is called first order, order 2 second order, etc.

Example: The differential equation  \large x \frac{d^2y}{dx^2}-5y=cx  is second order since the highest derivative is \large \frac{d^2y}{dx^2} or the second derivative.

                The Differential equation \large x \frac{dy}{dx}+25y=cx^3 is first order since the highest derivative is \large \frac{dy}{dx} or first derivatives only.

                     The Differential equation \large x \frac{d^3y}{dx^3}-6 \frac{dy}{dx}=cx^2is third order since the highest derivative is  \large \frac{d^3y}{dx^3} or third order derivatives only.                  

So the first concept is clear that the number of times the derivative of the function is taken is the Order of that differential equation

Lets move to the secon part that is Degree of the Differential Equation

Degree of Differential Equation

The Highest power of Highest order derivative in a differential equation is called the Degree of differential equation.

 This definition has one key point that if overlooked can result in wrong answers, i.e. You must look for the highest power of highest order derivative and not every derivative,

Let us see the example below

\large x \frac{d^3y}{dx^3}-6 ({\frac{dy}{dx}})^3=cx^2

Here the highest order = 3

and highest degree is ????? 3.....No it is wrong ...Because power 3 doesnot belongs to the highest order derivative in the equation, so it is not the degree of the equation,but actual degree of equation is 1. Why so ? Because the highest power of the third order (Highest order) Derivative is 1.

Answer is : Order:3 Degree:1

Let us try some more examples

ORDER = 3 and Degree=1  { Here 2 is not degree although it is highest in the equation because it does not belongs to highest order derivative..

Watch the videos below to get more clear idea and how to solve the examples....