In mathematics, a differential equation is an equation consisting of one or more functions together with their derivatives. Derivatives of a function define the rate of change of the function at a point. Differential equation is mainly used in physics, biology, economics, engineering and many others. The main objective of differential equations is the study of solutions that satisfy the properties of equations and solutions. Here we will discuss the differential equation definition, differential equation types, linear differential equation, methods of solving differential equation, exact differential equation, order and degree of differential equation, ordinary differential equation, and solution of differential equation.

Definition of Differential Equation

A differential equation is an equation consisting of one or more terms and the derivatives of one variable (i.e., the dependent variable) with respect to another variable (i.e., the independent variable).

Here “x” is the independent variable and “y” is the dependent variable.

For example, dy/dx = 5x

A differential equation has derivatives which are either partial derivatives or simple derivatives. The derivative represents the rate of change, and the differential equation describes the relationship between a quantity that varies continuously with respect to a change in some other quantity. There are many differential equation formulas for finding solutions to derivatives.

Order of Differential Equation

Order of a differential equation is the order of the higher order derivative present in equation. Here are some examples of different orders of differential equations.

dy/dx = 3x + 2 , The order of the equation is 1
(d2y/dx2)+ 2 (dy/dx)+y = 0. The order is 2
(dy/dt)+y = kt. The order is 1

First Order Differential Equation

As you can see in the first example, it is a first order differential equation with degree equal to 1. All linear equations in the form of derivatives are first order. It has only the first derivative as dy/dx, where x and y are two variables and is represented as:
dy/dx = f(x, y) = y’

Second-Order Differential Equation

An equation that involves a second order derivative is a second order differential equation. It is represented by;
d/dx(dy/dx) = d2y/dx2 = f”(x) = y”

Degree of Differential Equation

The degree of a differential equation is the power of the highest order derivative, where the original equation is represented as a polynomial equation in derivatives such that y’, y”, y”‘, and so on.

Let (d2y/dx2)+ 2 (dy/dx)+y = 0 be a differential equation, so the degree of this equation is 1. See some more examples here:

1. dy/dx + 1 = 0, degree is 1.
2. (y”’)3 + 3y” + 6y’ – 12 = 0, degree is 3
3. (dy/dx) + cos(dy/dx) = 0; it is not a polynomial equation in y′ and the degree of such a differential equation can not be defined.

Comment: The order and degree (if defined) of any differential equation are always positive integers.

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Types of Differential Equations

Differential equations can be divided into several types

• Ordinary Differential Equations
• Partial Differential Equations
• Linear Differential Equation
• Non linear Differential Equation
• Homogeneous Differential Equation
• Non-homogeneous Differential Equations

Ordinary Differential Equation

An ordinary differential equation consists of a function and its derivatives. It consists of only one independent variable and one or more of its derivatives with respect to the variables.

The order of ordinary differential equations is defined as the order of the highest derivative that occurs in the equation. The general form of an n-th order ODE is given by F(x, y, y’,…., yn ) = 0

Differential Equation Solution

A function which satisfies a given differential equation is called its solution. The solution in which the order of the differential equation has as many arbitrary constants is called a general solution. A solution free of arbitrary constants is called a special solution. There are two methods of finding the solution of a differential equation.

Separation of Variables

Separation of variables is done when the differential equation can be written in the form dy/dx = f(y)g(x), where f is a function of y only and g is a function of x only. Taking a starting position, rewrite the problem as 1/f(y)dy= g(x)dx and then integrate on both sides.

The Integrating Factor

The integrating factor technique is used when the differential equation is in the form dy/dx + p(x)y = q(x), where both p and q are functions of x.

The first-order differential equation is of the form y’ + P(x)y = Q(x). where P and Q are both functions of x and first derivatives of y. A higher-order differential equation is an equation that contains derivatives of an unknown function that can be either partial or normal derivatives. It can be shown in any order.

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Application

Differential equations have many applications in various fields such as applied mathematics, science, and engineering. Apart from technical applications, they are also used to solve many real life problems. Let’s look at some differential equation applications in real time.

1) Differential equations describe different exponential growth and decay.

2) They are also used to describe the change in return on investment over time.

3) They are used in the field of medical science for the development of cancer or the spread of disease in the body.

4) With its help, the speed of electricity can also be described.

5) They help economists in finding optimal investment strategies.

6) The motion of waves or pendulum can also be described using these equations.
There are various other applications in engineering: heat conduction analysis, in physics it can be used to understand the motion of waves. Ordinary differential equations can be used as an application in the engineering field to find the relationship between different parts of a bridge.

Now, let’s look at examples of differential equations in real life applications.

Linear Differential Equation Real World Examples

Let us consider this simple example to understand differential equations. Have you ever wondered why a hot cup of coffee gets cold when kept under normal conditions? According to Newton, the cooling of a hot body is proportional to the temperature difference between its temperature T and its surroundings T0. In mathematical terms, this statement can be written as: dT/dt ∝ (T – T0)…………(1)

It is in the form of a linear differential equation.

By introducing the proportionality constant k, the above equation can be written as:

dt / dt = k (t – t0) ……… (2)

Here, T is the body temperature and t is the time,

T0 is the ambient temperature,

dT/dt is the rate of cooling of the body

eg: dy/dx = 3x

Here, the differential equation has a derivative involving one variable (dependent variable, y) w.r.t another variable (independent variable, x). The types of differential equations are:

1. An ordinary differential equation consists of one independent variable and its derivatives. It is often called ODE. The general definition of an ordinary differential equation is of the form: given a F, a function os x and y and the derivative of y, we have

F(x, y, y’ …..y^(n1)) = y (n) is a categorical ordinary differential equation of order n.

2. Partial differential equation containing one or more independent variables.

Differential Equations Practice Questions

Question: Find the general solution of the differential equation dy/dx = 2x?

Solution:To solve this first-order differential equation, we integrate both sides with respect to x:

dy/dx = 2x

dy = 2x dx

Integration of both sides:

y = x^2 + C, where C is the constant of integration.

Therefore, the general solution is y = x^2 + C , where C is an arbitrary constant.

Question: Find the general solution of the differential equation y” + y = 0.

Solution: This is a second order homogeneous differential equation with constant coefficients. We assume a solution of the form y = e^(rt), where r is a constant.

Substituting this into the differential equation, we get:

r^2 e^(rt) + e^(rt) = 0

e^(rt)(r^2 + 1) = 0

Since e^(rt) is never zero, we must have r^2 + 1 = 0. Solving for r, we get r = ±i.

Therefore, the general solution is y = c1 cos(x) + c2 sin(x) , where c1 and c2 are arbitrary constants.

Question: Solve the initial value problem y’ = x^2, y(0) = 1?

Solution: We can solve this first-order differential equation by integrating both sides with respect to x:

dy/dx = x ^ 2

Integration of both sides:

y = (1/3) x^3 + c

Using the initial condition y(0) = 1, we can solve for C:

1 = (1/3) (0)^3 + c

c = 1

Therefore, the solution to the initial value problem is y = (1/3) x^3 + 1 .