Differential equations are the equations that consist of one or more functions along with their derivatives. The equations consist of derivatives of one variable which is called the dependent variable with respect to another variable which is called the independent variable. Differential equations may consist of one or more dependent and independent variables. Differential equations are widely used in various real-life problems. It is used in different domains of Mathematics, Physics, Chemistry, Economics, Biology, and Engineering. Differential equation definitions are also used to describe the exponential growth of things such as population, chemical decomposition etc, and in the computation of increase or decrease of a particular species of living organisms.

Differential equations are the equations that consist of a function along with its derivative of one variable called the dependent variable with respect to the independent

variable. If a general differential equation is represented as:

𝑓(𝒙) = \[\frac{{dy }}{dx}\]

In the above equation, the variable y is the dependent variable and the variable x is the independent variable. It means that the value of ‘y’ depends on the value of ‘x’. The derivative dy/dx can also be represented as y’ or f’(x).

A few examples of the differential equation are stated below:

\[\frac{{dy }}{dx}\] = 7x + 2

y' = y'' - 2

f(x) = 7 - f'(x) + x.f''(x)

Order and degree are the most important terms which should be very clearly understood while solving the differential equations. Order of a differential equation is the highest power to which the derivatives are raised in the given equation. However, degree on the other hand is the power of the highest derivative. For example, consider the differential equation mentioned below.

(y’)2 + y’’’ - 2 (y’’)4 = 7y

The equation consists of the third derivative of ‘y’ as y’’’ which is the highest derivative. The power of y’’’ is 1. So, the degree of the equation is ‘1’. However, the second derivative is y’’ which is raised to the power 4 which is the highest power of the derivative. So, the order of the given differential equation is 4.

Classification of differential equations is based on various factors such as type, order, linearity and homogeneity etc. The below chart summarizes the classification of differential equations.

Based on The Type of Variables, The Differential Equation Types Are:

Ordinary Differential Equations:

Ordinary differential equations are the equations that depend on only one independent variable.

Partial Differential Equations:

These are the differential equation types in which two or more independent variables affect the dependent variable.

Based on The Order of The Equations, The Differential Equation Types Are:

The order of the differential equations is the highest power of the derivative in that equation.

First order differential equation: The highest power of the derivative is one

Second order differential equation: The highest power of the derivative is two

Nth order differential equation: The highest power of the derivative may be any integer ‘n’.

Classification of Differential Equations Based on its Linearity is as follows:

Linear differential equations: These are the differential equations in which the power of the variable is always one.

Nonlinear differential equations: The differential equations in which the power of the variables in the equation is any number other than 1.

Based on Homogeneity, The Differential Equation Types Are:

Homogeneous differential equations

Non homogeneous differential equations

Solving a differential equation means to find an equation which does not contain any derivatives. However this equation should satisfy the differential equation that is being solved. Solving differential equations involves two or more integrations. To determine an appropriate method to solve the differential equation, it is very important to identify the type of differential equation that is being solved. Both general and particular solutions of differential equations can be obtained by using appropriate steps to solve the equation.

A differential equation will generally have an infinite number of solutions.

A general formula can be derived for the solution of a few differential equations.

FAQ (Frequently Asked Questions)

1. What is a differential equation? What are differential equation types?

Differential equations are the equations that consist of functions along with their derivatives. Differential equations usually consist of derivatives of one variable called the dependent variable with respect to another variable called the independent variable. Differential equations are classified into several types based on various parameters.

Based on the type of the variable used, they are classified as ordinary and partial differential equations.

Based on the order of differential equations, they are classified as first, second, third .. and nth order differential equations.

Linear and nonlinear differential equations

Homogeneous and nonhomogeneous differential equations

2. Where are differential equations used in real life?

Differential equations find an extensive range of applications in various domains of Science, Mathematics, Engineering and Technology. A few applications of differential equations include:

It is used to describe the functions which indicate the exponential growth and decay such as the growth of population and description of radioactive decay etc.

In economics, the differential equations are used to describe the return of investment with respect to time and also optimum investment strategies.

In biology, especially in the field of medicine, differential equations are also used to describe the growth or spread of diseases such as cancer in the human body.

Several physical quantities in Physics can be expressed in terms of differential equations. Differential equations are also used to describe the motion of waves and particles.