Differential Equation
A differential equation by definition is an equation that contains one or more functions with its derivatives. The rate of change of a function at a point is defined by the derivatives of the function. Differential equations are mainly used in the fields of biology, physics, engineering, and many. The main purpose of the differential equation is for studying the solutions that satisfy the equations and the properties of the solutions. In this article, we will learn how to solve a differential equation.
Explicit formulas help us to solve the differential equation. In this lesson, we shall discuss the definition, types, methods to solve the differential equation, order, and degree of the differential equation, types of differential equations, with realworld examples, and practice problems.
What are Differential Equations?
An equation that contains the derivative of a function is called a differential function. In calculus, a differential equation is an equation that involves the derivative (derivatives) of the dependent variable with respect to the independent variable (variables) is called a differential equation. A differential equation can contain derivatives that are either partial derivatives or ordinary derivatives. The derivative represents nothing but a rate of change, and the differential equation helps us present a relationship between the changing quantity with respect to the change in another quantity. Some of the differential equations formulas to find the solution of the derivatives is:
 (dy/dx) = sin x
 (d^{2}y/dx^{2}) + k^{2}y = 0
 (d^{2}y/dt^{2}) + (d^{2}x/dt^{2}) = x
 (d^{3}y/dx^{3}) + x(dy/dx)  4xy = 0
 (rdr/dθ) + cosθ = 5
Order of Differential Equations
The order of a differential equation is the highest order of the derivative appearing in the equation. Consider the following differential equations,
dy/dx = e^{x}, (d^{4}y/dx^{4}) + y = 0, (d^{3}y/dx^{3}) + x^{2}(d^{2}y/dx^{2}) = 0
In above differential equation examples, the highest derivative are of first, fourth and third order respectively.
First Order Differential Equation
You can see in the first example, it is a firstorder differential equation that has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
SecondOrder Differential Equation
The equation which includes secondorder derivative is the secondorder differential equation. It is represented as; d/dx(dy/dx) = d^{2}y/dx^{2} = f”(x) = y”
Degree of Differential Equations
The degree of the differential equation is the power of the highest ordered derivative present in the equation. To find the degree of the differential equation, we need to have a positive integer as the index of each derivative.
Types of Differential Equations
The different types of differential equations are:
 Ordinary Differential Equations
 Homogeneous Differential Equations
 Nonhomogenous Differential Equations
Ordinary Differential Equation
The “Ordinary Differential Equations” also known as ODE is an equation that contains only one independent variable and one or more of its derivatives with respect to the variable. Thus, the ordinary differential equation is represented as the relation having one independent variable x, the real dependent variable y, with some of its derivatives y’, y”, ….yn,…with respect to x.
Example: (d^{2}y/dx^{2}) + (dy/dx) = 3y cosx
The above differential equation example is an ordinary differential equation since it does not contain partial derivatives.
Homogenous Differential Equation
A differential equation in which the degree of all the terms is the same is known as a homogenous differential equation. In general they can be represented as P(x,y)dx + Q(x,y)dy = 0, where P(x,y) and Q(x,y) are homogeneous functions of the same degree.
Examples of Homogenous Differential Equation:
 y + x(dy/dx) = 0 is a homogenous differential equation of degree 1
 x^{4} + y^{4}(dy/dx) = 0 is a homogenous differential equation of degree 4
 xy(dy/dx) + y^{2} + 2x = 0 is not a homogenous differential equation
NonHomogenous Differential Equation
A differential equation in which the degree of all the terms is not the same is known as a homogenous differential equation.
Example: xy(dy/dx) + y^{2} + 2x = 0 is not a homogenous differential equation.
One of the types of a nonhomogenous differential equation is the linear differential equation, similar to the linear equation. The differential equation of the form (dy/dx) + Py = Q (Where P and Q are functions of x) is called a linear differential equation. (dy/dx) + Py = Q (Where P, Q are constant or functions of y). The general solution is y × (I.F.) = ∫Q(I.F.)dx + c where, I.F(integrating factor) = e^{∫pdx}
Solving Differential Equations
A solution that can be obtained from the general solution by giving particular values to arbitrary constants is called a particular solution. Therefore the differential equation has infinitely many solutions. Let us understand differential equation solver by one example
(dy/dx) = x^{2}y + y
Step 1: Divide the above differential equation by y.
(1/y)(dy/dx) = (x^{2} + 1)
We consider y and x both as variables and write this as
(dy/y) = (x^{2} + 1)dx
Step 2: Now integrate L.H.S. with respect to y and with respect to x.
∫(1/y)dx = ∫(x^{2} + 1)dx
Step 3: After integrating, we get:
log y = (x^{3}/3) + x + c
So, this is how the differential equation is solved.
Applications of Differential Equations
Ordinary differential equations applications in real life are used to calculate the movement or flow of electricity, motion of an object to and fro like a pendulum, to explain thermodynamics concepts. Also, in medical terms, they are used to check the growth of diseases in graphical representation.
Important Notes:
 We can use the following notations for derivatives. (dy/dx) = y', (d^{2}y/dx^{2}) = y'', (d^{3}y/dx^{3)} = y'''
 The order and degree of a differential equation should be always positive integers.
Also Read:
Differential Equations Examples

Example 1: What is the order of the differential equations (d^{2}y/dx^{2}) + x(dy/dx) + y = 2sinx?
Solution: The order of the given differential equation (d^{2}y/dx^{2}) + x(dy/dx) + y = 2sinx is 2.
Answer: The order is 2

Example 2: The rate of decay of the mass of a radio wave substance any time is k times its mass at that time, form the differential equations satisfied by the mass of the substance.
Solution: The rate of decay of mass is dm/dt.
Here dm/dt is directly proportional to m.
∴ dm/dt = mk (where k < 0) is the required differential equation.
Answer: ∴ dm/dt = mk is the required equation
FAQs on Differential Equation
What is a Differential Equation in Calculus?
An equation that contains the derivative of a function is called a differential function. A differential equation is an equation that involves the derivative (derivatives) of the dependent variable with respect to the independent variable (variables) is called a differential equation. Further, a differential equation contains derivatives of different orders and degrees.
What is the Formula of Differential Equation?
dy/dx = f(x); A differential equation contains derivatives which are either partial derivatives or ordinary derivatives. The derivative represents a rate of change, and the differential equation describes a relationship between the quantity that is continuously varying with respect to the change in another quantity.
What Are the Applications of Differential Equations?
Ordinary differential equations applications in real life are used to calculate the movement or flow of electricity, motion of an object to and fro like a pendulum, to explain thermodynamics concepts. Also, in medical terms, they are used to check the growth of diseases in graphical representation.
How Many Types of Differential Equations Are There?
The different types of differential equations are:
 Ordinary Differential Equations
 Homogeneous Differential Equations
 Nonhomogeneous Differential Equations
 Linear Differential Equations
 Nonlinear Differential Equations
What are Homogeneous Differential Equations?
A differential equation in which the degrees of all the terms is the same is known as a homogenous differential equation. x^{2} + y^{2}xy and xy + yx are examples of homogenous differential equations. y + x(dy/dx) = 0 is a homogenous differential equation of degree 1. x^{4} + y^{4}(dy/dx) = 0 is a homogenous differential equation of degree 4.
What are Second Order Differential Equations?
A secondorder differential equation is one in which there is a second derivative but not a third or higher derivative. It is represented as d/dx(dy/dx) = d^{2}y/dx^{2} = f”(x) = y”
How to Find Order and Degree of Differential Equations?
The order of a differential equation is the highest order of the derivative appearing in the equation. The degree of the differential equation is the exponent of the highest ordered derivative present in the equation. Consider the following differential equations, dy/dx=e^{x}, d^{4}y/dx^{4 }+ y = 0, (d^{3}y/dx^{3}) + x^{2}(d^{2}y/dx^{2}) = 0, in the given differential equation examples, the highest derivative are of first, fourth and thirdorder respectively.