Differential Equation

What is Differential Equation?

A differential equation is a mathematical equation that involves one or more functions and their derivatives. The rate of change of a function at a point is defined by its derivatives. It's mostly used in fields like physics, engineering, and biology. The analysis of solutions that satisfy the equations and the properties of the solutions is the primary goal of differential equations. Using explicit formulas is one of the simplest ways to solve the differential equation. In this article, let us discuss the differential equation meaning, types, methods to solve the differential equation, order and degree of the differential equation, differential equations formulas and few solved problems.


Differential Equation Definition

A differential equation has one or more terms as well as the derivatives of one variable (the dependent variable) in relation to another variable (i.e., independent variable)

\[\frac{dy}{dx}\] = f(x)

Here "x" is the independent variable, and "y" is the dependent variable.

For example, \[\frac{dy}{dx}\] = 5x

Partially derivatives and ordinary derivatives are also present in a differential equation. The differential equation defines a relationship between a quantity that is continuously varying with respect to a change in another quantity, and the derivative represents a rate of change.


What is a Differential Equation?

An equation involving an unknown function y=f(x) and one or more of its derivatives is known as a differential equation. To put it another way, it's an equation that involves derivatives of one or more dependent variables with respect to one or more independent variables. Differential equations are important for describing nature mathematically, and they are at the heart of many physical theories.


Order of Differential Equation

The order of differential equation is the order of the equation's highest order derivative present in the equation. Here are some examples of differential equations in various orders.

\[\frac{d^{3}x}{dx^{3}}\] + 3x\[\frac{dy}{dx}\] = e\[^{y}\]

The order of the highest derivative in this equation is 3, indicating that it is a third-order differential equation.

Example - (\[\frac{d^{2}y}{dx^{2}}\])\[^{4}\] + \[\frac{dy}{dx}\] = 3

This equation represents a second-order differential equation.

This way we can have higher-order differential equations i.e n\[^{th}\] order differential equations.

 

First Order Differential Equation

As you can see in the first example, the differential equation is a First Order Differential Equation with a degree of 1. In the form of derivatives, all linear equations are in the first order. It only has the first derivative, dy/dx, where x and y are the two variables, and is written as:

dy/dx = f(x, y) = y’

For example \[\frac{dy}{dx}\] + (x\[^{2}\] + 5)y = \[\frac{x}{5}\]

This also represents a First-order Differential Equation.

 

Second-Order Differential Equation

The equation which includes second-order derivatives is the second-order differential equation.  It is represented as;

\[\frac{d}{dx}\](\[\frac{dy}{dx}\]) = \[\frac{d^{2}y}{dx^{2}}\] = f’’(x) = y’’

For example \[\frac{d^{2}y}{dx^{2}}\] + (x\[^{3}\] + 3x)y = 9 

In this case, the highest derivative's order is 2. As a result, the equation is a second-order differential equation.

 

Types of Differential Equations

Differential equations are classified into many categories. They are:

  • Ordinary  Differential Equations

  • Partial  Differential Equations

  • Linear Differential Equations

  • Non-linear Differential Equations

  • Homogeneous Differential Equations

  • Non-homogeneous  Differential Equations

Differential Equations Solutions

The solution to the differential equation can be found using one of two methods.

  • Separation of variables

  • Integrating factor

The variable is isolated when the differential equation can be written in the form dy/dx = f(y)g(x), where f is the function of y only and g is the function of x only. Rewrite the problem as 1/f(y)dy= g(x)dx and then integrate on both sides using an initial condition.


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

 

y'+ P(x)y = Q is a first-order differential equation (x). P and Q are functions of x and the first derivative of y, respectively. An equation containing partial or ordinary derivatives of an unknown function is referred to as a higher-order differential equation. It is possible to represent it in any order.

 

Degree of Differential Equation

The power of the highest order derivative, where the original equation is defined as a polynomial equation in derivatives such as y',y”, y”', and so on, is the Degree of Differential Equation.

Assume that \[\frac{d^{2}y}{dx^{2}}\] + 2(\[\frac{dy}{dx}\]) + y = 0 is a differential equation, in which case the degree of this equation is 1. Here are some more examples:

dy/dx + 1 = 0, degree is 1

(y”’)3 + 3y” + 6y’ – 12 = 0, in this equation, the degree is 3.

 

Ordinary Differential Equation

The function and its derivatives are involved in an ordinary differential equation. Just one independent variable and one or more of its derivatives with respect to the variable are used.

Ordinary differential equations have an order that is defined as the order of the highest derivative in the equation. The n\[^{th}\] order ODE's general form is as follows:

F(x, y, y’,…., yn ) = 0

2\[\frac{d^{2}y}{dx^{2}}\] + (\[\frac{dy}{dx}\])\[^{3}\] = 0 is an ordinary differential equation.

 

Linear Differential Equations

A differential equation of the form: \[\frac{dy}{dx}\] + My = N 

The first-order linear differential equation, where M and N are constants or functions of x only, The following is an example of first-order linear differential equations: \[\frac{dy}{dx}\] + y = sinx

 

Linear Differential Equations Real World Example

Find this basic example to better understand differential equations. Have you ever wondered why a hot cup of coffee cools down when held at room temperature? A hot body's cooling is proportional to the temperature difference between its temperature T and the temperature T\[_{0}\] of its surroundings, according to Newton. In terms of mathematics, this sentence can be written as:

dT/dt ∝ (T – T\[_{0}\])…………(1)

A linear differential equation takes this shape.

With the addition of a proportionality constant k, the above equation becomes:

dT/dt = k(T – T\[_{0}\])  …………(2)

T is the body temperature, and t is the time in this equation.

T\[_{0}\] is the temperature of the surrounding,

The rate of cooling of the body is dT/dt.

 

Applications of Differential Equations:

1) Differential equations are used to explain the growth and decay of various exponential functions.

2) They can also be used to explain how a return on investment changes over time.

3) They're used in medical science to model cancer growth and disease spread across the body.

4) It may also be used to explain the movement of electricity.

5) They assist economists in determining the most effective investment strategies.

6) These equations may also be used to explain the motion of waves or a pendulum.

 

Differential Equations Examples

1. Form the Differential Equation y=mx, Where m is an Arbitrary Constant, to Describe the Family of Curves.

Sol: Here we will eliminate constant by differentiating y,

Given y=mx

First, calculate the value of \[\frac{du}{dx}\] 

\[\frac{dy}{dx}\] = \[\frac{d(mx)}{dx}\]  

⇒ \[\frac{dy}{dx}\] = m

Or m = \[\frac{dy}{dx}\]

Now, the given equation is y=mx

Put the value of m in the above equation,

y = \[\frac{dy}{dx}\](x)

⇒ x(\[\frac{dy}{dx}\]) - y = 0

Hence differential equation is x(\[\frac{dy}{dx}\]) - y = 0


2. Determine the Order and Degree of the Differential Equation 2x\[^{2}\]\[\frac{d^{2}y}{dx^{2}}\] - 3\[\frac{dy}{dx}\] + y = 0 is

Sol: Given equation is  2x\[^{2}\]\[\frac{d^{2}y}{dx^{2}}\] - 3\[\frac{dy}{dx}\] + y = 0

We can write it as 2x\[^{2}\]y’’ - 3y’ + y = 0 

The highest order of the derivative is 2 and the degree of the highest derivative is 1 as shown in the equation.

Hence order is 2 and degree 1 of the differential equation 2x\[^{2}\]\[\frac{d^{2}y}{dx^{2}}\] - 3\[\frac{dy}{dx}\] + y = 0


3. The Integrating Factor of the Differential Equation x\[\frac{dy}{dx}\] - y = 2x\[^{2}\] is

Sol: Given equation x\[\frac{dy}{dx}\] - y = 2x\[^{2}\] 

Divide both sides by x,

⇒ \[\frac{dy}{dx}\] - \[\frac{y}{x}\] = 2x

Now compare the above equation with the standard equation of the form, 

\[\frac{dy}{dx}\] + Py = Q

Here P = -  \[\frac{1}{x}\] and Q = 2x

So Integrating Factor (IF) = e\[^{\int pdx}\] 

IF = e\[^{\int -\frac{1}{x}dx}\]

IF = e\[^{-logx}\] 

IF = e\[^{log x^{-1}}\]

IF = x\[^{-1}\]

IF = \[\frac{1}{x}\]

Hence integrating factor = \[\frac{1}{x}\]


Conclusion:

From above discussion we can say a differential equation is defined as an equation containing the derivative or derivatives of the dependent variable with respect to the independent variable. We have discussed differential equations formulas and different methods to solve differential equations by using basic differential equations formulas. 

FAQs (Frequently Asked Questions)

1. What is the Differential Equation?

Ans: A differential equation is a mathematical equation that has one or more derivatives of a function. The function's derivative is given by dy/dx. A solution to a differential equation is a function y=f(x) that contains the differential equation when f and its derivatives are substituted into the equation.

2. What is the Use of a Differential Equation?

Ans: The differential equation's main aim is to calculate the function over its entire domain. It's a term for the exponential growth or decay of a system over time. It has the power to foresee what will happen in the world around us. It is commonly used in a variety of areas, including physics, chemistry, biology, and economics.

3. What is the General Solution of Differential Equations?

Ans: In the case of an ODE, the general solution contains all possible solutions and usually includes arbitrary constants or arbitrary functions (in the case of a PDE.) A specific solution does not contain any arbitrary constants or functions.