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

What is Differential Equation?

A differential equation is an equation with one or more variables (unknowns) and some of their derivatives. That means the differential equation defines the relationship between variables and their derivatives.

In this article, you will learn the definition of the order and degree of differential equations and how to find the order and degree of a given differential equation, along with solved examples.

Order of Differential Equation

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

Example (i):

\(\begin{array}{l}\frac{d^3 x}{dx^3} + 3x\frac{dy}{dx} = e^y\end{array} \)

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

Example (ii): 

\(\begin{array}{l} (\frac{d^2 y}{dx^2})^ 4 + \frac{dy}{dx}= 3 \end{array} \)

This equation represents a second order differential equation.

This way we can have higher order differential equations i.e., nth order differential equations.

Differential-Equation

First order differential equation

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

\(\begin{array}{l}\frac{dy}{dx} + Py = Q \end{array} \)

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

This represents a linear differential equation whose order is 1.

Example:

\(\begin{array}{l} \frac{dy}{dx} + (x^2 + 5)y = \frac{x}{5} \end{array} \)

This also represents a First order Differential Equation.

Learn more about first order differential equations here.

Second Order Differential Equation

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

Example:

\(\begin{array}{l}\frac{d^2 y}{dx^2} + (x^3 + 3x) y = 9 \end{array} \)

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 the 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: 

\(\begin{array}{l}\frac{d^4 y}{dx^4} + (\frac{d^2 y}{dx^2})^2 – 3\frac{dy}{dx} + y = 9 \end{array} \)

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:

\(\begin{array}{l} [\frac{d^2 y}{dx^2} + (\frac{dy}{dx})^2]^4 = k^2(\frac{d^3 y}{dx^3})^2\end{array} \)

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 the highest derivative is 2.

Special Case (Degree is Not Defined)

When is the Degree of Differential Equation is not Defined?

It is not possible every time that we can find the degree of a given differential equation. The degree of any differential equation can be found when it is in the form of a polynomial; otherwise, the degree cannot be defined.

Suppose in a differential equation dy/dx = tan (x + y), the degree is 1, whereas for a differential equation tan (dy/dx) = x + y, the degree is not defined. These types of differential equations can be observed with other trigonometry functions such as sine, cosine and so on.

Let us see some more examples of finding the degree and order of differential equations.

Solved Questions

Question 1:– Write the degree and order of DE:

\(\begin{array}{l}\frac{d^2 y}{dx^2} + cos\frac{d^2 y}{dx^2} = 5x\end{array} \)

Solution:

Given DE: 

\(\begin{array}{l}\frac{d^2 y}{dx^2} + cos\frac{d^2 y}{dx^2} = 5x\end{array} \)

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

Question 2: Find the order and degree of the differential equation:

\(\begin{array}{l}(\frac{d^3 y}{dx^3})^2 + y = 0\end{array} \)

Solution:

Given,

\(\begin{array}{l}(\frac{d^3 y}{dx^3})^2 + y = 0\end{array} \)

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

Question 3: Figure out the order and degree of differential equation that can be formed from the equation:

\(\begin{array}{l}\sqrt{1 – x^2} + \sqrt{1 – y^2} = k(x – y)\end{array} \)

Solution:

\(\begin{array}{l}Let\ x = sin \theta, y = sin \phi \end{array} \)

So, the given equation can be rewritten as

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

\(\begin{array}{l} \Rightarrow (cos \theta + cos \phi) = k(sin \theta – sin \phi)\end{array} \)

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

\(\begin{array}{l}cot \frac{\theta – \phi}{2} = k\end{array} \)

\(\begin{array}{l}\theta – \phi = 2cot^{-1} k\end{array} \)

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

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

\(\begin{array}{l}\frac{1}{\sqrt{1 – x^2}} – \frac{1}{\sqrt{1 – y^2}}\, \frac{dy}{dx} = 0\end{array} \)

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

Note: If the DE in which differential coefficient is present inside the parenthesis of any another function as a composite, then first attempt to make it as simple as possible. Now, check whether it is in the form of a polynomial in terms of derivatives. If it is a polynomial, the degree can be defined.

Practice Problems

  1. Find the degree and order of differential equation dy/dx + sin x = 0.
  2. What is the order of the differential equation (d3y/dx3) – 2y(dy/dx) + 4 = 0?
  3. Identify the degree and order for the differential equation (d3y/dx3) + 4(d2y/dx2)2 + (dy/dx) = 0.

Related Articles

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.

Frequently Asked Questions – FAQs

How do you define the order of a differential equation?

The order of a differential equation is the highest order of the derivative of a variable in the given differential equation.

What is the degree of a differential equation?

The degree of a differential equation is the power of the highest-order derivative.

Is it possible to define the degree of all differential equations?

No, we cannot define the degree of all differential equations. To write the degree, the given differential equation must be in the form of a polynomial.

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