## 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 order 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 fourth order differential equation.

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

### 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

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

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 Byjuâ€™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.