Differential Equations

Introduction

We need to develop various mathematical models to establish relationships between multiple variables in real life. In this direction, differential equations play an important role. These are applied parts of mathematics and used in calculus. In this tutorial, we will discuss the meaning, order, degree, and types of differential equations with solved examples.

Differential Equations

Differential equations are mathematical statements containing functions and their derivatives.

It describes the relationship between the variables with their rate of change. It is used in engineering, science, biology, finance, etc.

The differential equations contain at least one ordinary derivative or partial derivative term.

Let’s consider the rate of change of a variable p with respect to x is directly proportional to x. The given statement can be written in the form of a differential equation as

$\mathrm{\frac{dp}{dx}\:=\:kx\:(Where\:k\:is\:a\:constant)}$

Moreover, some examples of differential equations are listed below.

• $\mathrm{\frac{dp}{dx}\:=\:\sin^{2}x}$

• $\mathrm{\frac{d^{2}p}{dx^{x}}\:-\:2x\:=\:0}$

• $\mathrm{\frac{d^{2}p}{dx^{2}}\:-\:\frac{d^{2}q}{dx^{2}}\:+\:p^{2}\:+\:x^{2}\:=\:0}$

• $\mathrm{\frac{dp}{dx}\:+\:p^{2}\:=\:10}$

Order and Degree

Two important terms are associated with the differential equations, including order and degree. Let’s discuss each term in detail.

Order

The order of the differential equations can be defined as the maximum order derivative present in the statement. The differential equations are called by their order. The order may be any number like one, two, three, or four. Let’s discuss some examples.

First-order differential equations

If the order of the differential equation is one, it is known as a first-order differential equation. These equations are usually written in linear form. It contains the first derivative of a function. Some examples of first-order differential equations are listed below.

• $\mathrm{\frac{dp}{dx}\:+\:7x\:=\:1}$

• $\mathrm{\frac{dp}{dx}\:-\:\cos\:x\:=\:0}$

• $\mathrm{\frac{dp}{dx}\:=\:e^{2x}}$

Second-order differential equations

If the order of the differential equation is two, it is known as a second-order differential equation. It contains the second derivative of the function. Some examples of the second-order differential equations are listed below.

• $\mathrm{\frac{d^{2}p}{dx^{x}}\:+\:1\:=\:0}$

• $\mathrm{\frac{d^{2}p}{dx^{2}}\:+\:\frac{1}{x^{2}}\:=\:4}$

• $\mathrm{\frac{d^{2}p}{dx^{2}}\:+\:e^{2x}\:=\:2}$

Degree

The degree of a differential equation is defined as the exponent or power to which the highest derivative is raised. To find the degree of a differential equation, we need to express the statement as a polynomial equation. If the statement cannot be expressed in the form of polynomial expression, then there will be no degree for the differential equation.

Let’s consider an example of a differential equation,

$\mathrm{(\frac{d^{2}p}{dx^{2}})^{3}\:+\:\frac{dp}{dx}\:-\:5\:=\:0}$

In this case, the degree of the differential equation is 3.

Types of Differential Equations

There are several type of differential equations that are described below.

• Ordinary differential equations

• Partial differential equations

• Linear differential equations

• Non-linear differential equations

• Homogenous differential equations

• Non Homogeneous differential equations

Let’s discuss each type of differential equation in detail

Ordinary differential equations

The ordinary differential equations contain one or multiple functions of an independent variable and its derivative term. It is abbreviated as ODE. Some examples of ordinary differential equations are listed below.

• $\mathrm{\frac{dp}{dx}\:+\:7x^{4}\:=\:-8}$

• $\mathrm{\frac{dp}{dx}\:=\:x\:-\:y}$

• $\mathrm{\frac{d^{2}p}{dx^{2}}\:+\:\frac{dp}{dx}\:-\:x\:=\:5}$

Partial differential equations

The partial differential equations contain multiple independent variables, adependent variable, a partial derivative of the dependent variable with respect to the independent variable. It is abbreviated as PDE. Some examples of partial differential equations are listed below

• $\mathrm{\frac{\partial^{2}p}{\partial^{2}x}\:+\:5px\:=\:0}$

• $\mathrm{\frac{\partial^{2}p}{\partial^{2}x}\:+\:\frac{\partial^{2}q}{\partial^{2}x}\:-\:4p^{2}\:=\:0}$

Linear Differential equation

A linear polynomial equation containing derivatives of several terms is known as a linear differential equation.

• $\mathrm{\frac{dp}{dx}\:+\:2x^{2}y\:=\:-1}$

• $\mathrm{\frac{dp}{dx}\:-\:x\:=\:-8}$

Non-linear Differential equation

A differential equation that can not be expressed as a linear polynomial form is known as a nonlinear differential equation

• $\mathrm{\frac{dp}{dx}\:=\:p^{\frac{1}{3}}}$

• $\mathrm{\frac{dp}{dx}\:-\:6x\:=\:2p^{4}}$

Homogeneous Differential equation

If a differential equation 𝑓(𝑥, 𝑦) can be expressed as $\mathrm{m^{n}\:g(x\:,\:y)}$, then it is called a homogeneous differential equation

• $\mathrm{\frac{dp}{dx}\:=\:\frac{x\:+\:5}{x\:-\:5}}$

• $\mathrm{\frac{dp}{dx}\:=\:\frac{x^{2}}{x^{2}\:-\:4}}$

Non-homogeneous Differential equation

If a differential equation 𝑓(𝑥, 𝑦) cannnot be expressed as $\mathrm{m^{n}\:g(x\:,\:y)}$, then it is called a non-homogeneous differential equation.

• $\mathrm{\frac{dp}{dx}\:=\:\frac{x^{2}\:-\:2}{x}}$

Solved Examples

Example 1

Determine the order and degree of the following differential equation −

• $\mathrm{(\frac{d^{2}p}{dx^{3}})^{2}\:+\:(\frac{dp}{dx})^{4}\:+\:2x\:=\:0}$

Solution −

The maximum order derivative that is present in the given differential equation is 3.

Therefore the order of the equation is 3

Similarly, the degree is the exponent by which the highest derivative is raised. In this case, the degree is 2.

∴ The order and degree of the given differential equation are 3 and 2, respectively

Example 2

State which of the following differential equations are linear

• $\mathrm{\frac{dp}{dx}\:+\:x^{3}y\:=\:5}$

• $\mathrm{\frac{d^{2}p}{dx^{2}}\:+\:lnp\:-\:6\:=\:0}$

• $\mathrm{\frac{dp}{dx}\:+\:y^{5}\:=\:0}$

Solution −

• Since the term $\mathrm{\frac{dp}{dx}}$ and y are linear; hence the first equation is a linear differential equation.

• lnp is not a linear term. Therefore, the second equation is not a linear differential equation.

• $\mathrm{y^{5}}$ is not a linear term. Therefore, the second equation is not a linear differential equation

Example 3

Check whether the unction $\mathrm{p\:=\:2x^{2}}$ is the solution of $\mathrm{\frac{d^{2}p}{dx^{2}}\:+\:\frac{dp}{dx}\:+\:\frac{dp}{dx}\:-\:4\:=\:0}$

The given solution is $\mathrm{p\:=\:2x^{2}}$

Now, take differentiation on both sides.

$$\mathrm{\frac{dp}{dx}\:=\:\frac{d}{dx}\:(2x^{2})\:=\:4x}$$

Again take differentiation on both sides.

$\mathrm{\frac{dp}{dx}\:=\:\frac{d}{dx}\:(2x^{2})\:=\:4x}$

Now, put the values of $\mathrm{p\:,\:\frac{dp}{dx}\:and\:\frac{d^{2}p}{dx^{2}}}$ in the given differential equation.

$\mathrm{LHS\:=\:\frac{d^{2}p}{dx^{2}}\:+\:\frac{dp}{dx}\:-\:4\:=\:4\:+\:4x\:-\:4\:=\:\:4x\:\neq\:0}$

Hence,$\mathrm{LHS\:\neq\:RHS}$

∴ The given function is not a solution of $\mathrm{\frac{d^{2}p}{dx^{2}}\:+\:\frac{dp}{dx}\:-\:4\:=\:0}$

Word problems

Problem 1 − Determine the order and degree of the following differential equation

• $\mathrm{(\frac{d^{2}p}{dx^{2}})^{\frac{1}{2}}\:+\:(\frac{dp}{dx})^{2}\:+\:2x\:=\:0}$

• $\mathrm{(\frac{d^{2}p}{dx^{3}})^{4}\:-\:(\frac{dp}{dx})^{5}\:-\:7x^{2}\:=\:0}$

Problem 2 − Check whether the unction 𝑝 = 𝑙𝑛𝑥 is the solution of $\mathrm{(\frac{d^{2}p}{dx^{2}})^{2}\:-\:5\:\frac{dp}{dx}\:=\:0}$

Problem 3 − The number of bacteria in a culture at any time is m times the initial population. Formulate the differential equation.

Conclusion

The present tutorial gives a brief introduction about differential equations. In addition, some basic terms associated with the differential equations have been briefly discussed. Moreover, some solved examples have been provided for better clarity of this concept. In conclusion, the present tutorial may be useful for understanding the basic concept of differential equations

FAQs

1. How many solutions can have a differential equation?

There may be more than one solution for a differential equation.

2. What are the applications of differential equations?

These are used in various fields of engineering, science, biology, finance, etc. In specific, they are utilised to solve problems related to the flow of electricity, motion of an object, growth of microorganisms, and thermodynamic concepts.

3. How to find the solution of a differential equation?

There are two methods used in calculus to determine the solution of a differential equation.

• Separation of variables

• Integrating factor

We will do a detailed analysis of these methods in the next tutorial.

4. What do you mean by the exact differential equation?

A first-order differential equation is said to be an exact differential equation if it results in a simple differentiation.

5. What is the difference between calculus and differential equations?

Calculus is a subsection of mathematics that deals with differentiation and integration. The differential equation is a type of calculus dealing with the variables and their derivative terms.