Differential Equations: Definition, Types, Formula

Differential Equations: Differential Equation is an equation that involves the independent variable and the derivatives of the dependent variable. It represents the physical quantities and rate of change of a function at a point and is used in the field of Mathematics, Engineering, Physics, Biology and so on. Its prime focus is to study the solutions that satisfies each equation and the properties of their solutions.

Newton and Leibniz brought differential equation into existence. Ordinary, Partial, Linear, Non-Linear, Homogenous and Non-Homogenous differential equation and some of the types of Differential Equations. Students learn these equations in their secondary class in order to solve the mathematical problems easily. You can check NCERT Solutions for Class 12 Maths, Chapter 9 for better understanding of the concept. We have provided detailed information on differential equation in this article. Read on to find out about its definition, types, formula and examples.

Definition of Differential Equations

It is an equation that involves derivatives of the dependent variable with respect to independent variable. An equation of the form is known as Differential equation. The equation is related with one or more function and its derivatives. They are either ordinary or partial derivatives. These equations are used in various field of Engineering, Mathematics, Physics, Chemistry, Biology, Anthropology, Geology, Economic etc. Hence, it is considered to be important in all modern scientific investigations.

Formula of Differential Equations

The formula used in calculation of differential equation is mentioned below:

Where, f is the function and y is a derivative.

Types of Differential Equations

Differential equations are divided into several types such as:

1. Ordinary Differential Equation
2. Partial Differential Equation
3. Linear Differential Equation
4. Non-linear Differential Equation
5. Homogenous Differential Equation
6. Non-homogenous Differential Equation

Some of the basic concepts of differential equation are mentioned below:

1. The Ordinary differential equation involves derivatives of the dependent variable with respect to only one independent variable. The formula of ordinary differential equation is F(x, y, y’, …., y^n-1) = yⁿ
2. The partial differential equation involves derivatives with respect to more than one independent variables. The formula of partial differential equation is ∂u/∂x (x,y) = 0. 
3. The Order of a differential equation is defined as the order of the highest order derivative of dependent variable with respect to the independent variable.
4. A solution is defined as a function which satisfies the given differential equation.
5. First order linear differential equation is a equation of the form dy/dx +Py Q, where P and Q are constants or functions of x only.

Order of Differential Equation

It is the highest order derivatives of the dependent variable in an equation. There are different orders in differential equations as mentioned below:

First order differential equation is the equation which has degree equal to one. It represented as:

dy/dx = f(x, y) = y’

Where, dy/dx first order derivative and x and y are two variables.

Second order of differential equation is the equation which includes second order derivatives. It is represented as:

d/dx(dy /dx) = d²y/dx² = f”(x) = y”

Degree of Differential Equation

Degree of a differential equation is the power of the highest order derivative. To find the degree we represent original equation in the form of a polynomial equation in derivatives such as y’,y”, y”’.

For example, If (d²y/dx²)+ 2 (dy/dx)+y = 0  is a differential equation, then in polynomial equation form it is written as: y” + 2 y’ + y = 0. Since the highest order derivative y” has power 1 so its degree is 1.

Examples of Differential Equations

Some of the examples of differential equation are given below:

Example 1: Determine order and degree (if defined) of differential equation ? 4? /?? 4 + sin(? ′′′) = 0
Solution: Given differential equation is ? 4? /?? 4 + sin(? ′′′) = 0
We know that the highest order derivative present in the differential equation is 4. Therefore, its order is four. The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

Example 2: Determine order and degree (if defined) of differential equation ? ′ + 5? = 0
Solution: The given differential equation is ? ′ + 5? = 0. The highest order derivative present in the differential equation is ?′. Therefore, its order is one. It is a polynomial equation in its derivatives and the highest power raised to ?′ is 1. Hence, its degree is one.

Also check,

FAQs on Differential Equation

The frequently asked questions on differential equation are given below:

Q. What is differential equation?
Ans: It is an equation that involves derivatives of the dependent variable with respect to independent variable.

Q. What is order of differential equation?
Ans: The Order of a differential equation is defined as the order of the highest order derivative of dependent variable with respect to the independent variable.

Q. What is the formula of differential equation?
Ans: The formula of differential equation is dy/dx = f(x) Where, f is the function and y is the derivative.

Q. What are the types of differential equation?
Ans: The types of differential equation are ordinary, partial, linear, non-linear, homogenous and non-homogenous.

Q. What is the number of arbitrary constants in the general solution of a differential equation of fourth order?
Ans: The number of constants in the general solution of a differential equation of order n is equal to its order. Hence, the number of constants in the general equation of the fourth-order differential equation is four.

Q. Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants.
Ans: The required differential equation is as below:

Given,
y = a sin (x + b) … (1)
Differentiating both sides of equation (1) with respect to x,
dy/dx = a cos (x + b) … (2)
Differentiating again on both sides with respect to x,
d²y/dx² = – a sin (x + b) … (3)
Eliminating a and b from equations (1), (2) and (3),
d²y/ dx² + y = 0 … (4)
The above equation is from the arbitrary constants a and b.

We have provided detailed information on differential equations in this article. To prepare for the exam, Embibe provides CBSE study material that covers the whole CBSE Class 12 syllabus for Maths. You can also solve Maths practice questions for every chapter in the CBSE Class 12 syllabus for Maths that will also help you in your preparation.

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