## Order of Differential Equation:-

Differential Equations are classified on the basis of the order. Order of a differential equation is the order of the highest derivative (also known as differential coefficient) present in the equation.

For Example (i): \(\frac{d^3 x}{dx^3} + 3x\frac{dy}{dx} = e^y\)

In this equation the order of the highest derivative is 3 hence this is a third order differential equation.

Example (ii) : –\( (\frac{d^2 y}{dx^2})^ 4 + \frac{dy}{dx}= 3 \)

This equation represents a second order differential equation.

This way we can have higher order differential equations i.e. \( n^{th}\) order differential equations.

### First order differential equation:

The order of highest derivative in case of first order differential equations is Â 1. A linear differential equation has order 1. In case of linear differential equations, the first derivative is the highest order derivative.

\(\frac{dy}{dx} + Py = Q \)

P and Q are either constants or functions of the independent variable only.

This represents a linear differential equation whose order is 1.

Example: \( \frac{dy}{dx} + (x^2 + 5)y = \frac{x}{5} \)

This also represents a First order Differential Equation.

### Second Order Differential Equation:

When the order of the highest derivative present is 2, then it represents a second order differential equation.

Example: \(\frac{d^2 y}{dx^2} + (x^3 + 3x) y = 9 \)

In this example, the order of the highest derivative is 2. Therefore, it is a second order differential equation.

### Degree of Differential Equation:

The degree of differential equation is represented by the power of the highest order derivative in the given differential equation.

The differential equation must be a polynomial equation in derivatives for the degree to be defined.

Example 1:- \(\frac{d^4 y}{dx^4} + (\frac{d^2 y}{dx^2})^2 – 3\frac{dy}{dx} + y = 9 \)

Here, the exponent of the highest order derivative is one and the given differential equation is a polynomial equation in derivatives. Hence, the degree of this equation is 1.

Example 2: \( [\frac{d^2 y}{dx^2} + (\frac{dy}{dx})^2]^4 = k^2(\frac{d^3 y}{dx^3})^2\)

The order of this equation is 3 and the degree is 2 as the highest derivative is of order 3 and the exponent raised to highest derivative is 2.

Example 3:- \(\frac{d^2 y}{dx^2} + cos\frac{d^2 y}{dx^2} = 5x\)

The given differential equation is not a polynomial equation in derivatives. Hence, the degree for this equation is not defined.

Example 4:- \((\frac{d^3 y}{dx^3})^2 + y = 0\)

The order of this equation is 3 and the degree is 2.

Example 5:- Figure out the order and degree of differential equation that can be formed from the equation \(\sqrt{1 – x^2} + \sqrt{1 – y^2} = k(x – y)\).

Solution:-

Let \(x = sin \theta, y = sin \phi \)

So, the given equation can be rewritten as

\(\sqrt{1 – sin\theta^2} + \sqrt{1 – sin\phi^2} = k(sin \theta – sin \phi)\)

\( \Rightarrow (cos \theta + cos \phi) = k(sin \theta – sin \phi)\)

\( \Rightarrow 2 cos \frac{\theta + \phi}{2} cos\frac{\theta – \phi}{2} = 2 k cos \frac{\theta + \phi}{2} sin \frac{\theta – \phi}{2} \)

\(cot \frac{\theta – \phi}{2} = k\)

\(\theta – \phi = 2cot^{-1} k\)

\( sin^{-1}x – sin^{-1}y = 2cot^{-1} k\)

Differentiating both sides w. r. t. x, we get

\(\frac{1}{1 – x^2} – \frac{1}{1 – y^2}\, \frac{dy}{dx} = 0\)

So, the degree of the differential equation is 1 and it is a first order differential equation.

In the upcoming discussions, we will learn about solutions to the various forms of differential equations. We here at BYJU’S will help you tackle all your doubts in the easiest possible way. Visit us to enjoy the beauty of simplicity in solving all your doubts.