# Differential Equations

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## What Are Differential Equations?

Because equations may be used to simulate a wide range of real-world scenarios, differential equations are significant in both mathematics and the sciences. Leibniz, the Bernoulli brothers, and others first developed differential equations in the 1680s, not long after Newton developed "fluxional equations" in the 1670s. Specifically, differential equations as a separate discipline of mathematics began in 1693 when Leibniz and Newton both eventually formally published and distributed solutions to their differential questions.

Differential equations are used to simulate the behavior of complicated systems in both biology and economics. The sciences, from which the equations originated and where the solutions found application, were the first to influence the mathematical theory of differential equations. In this article, we will introduce the definition, examples, types, and applications of differential equations.

### Definition

An equation containing the derivative of the dependent variable (unknown function) with respect to the independent variable is called a differential equation. It is denoted by (DE).

### Example

Dy/Dx+9 = 0

In above example,

D is a differential.

y is Dependent variable which treat as unknown variable.

x is an independent variable.

9 is a constant.

### Types of Differential Equations

There are two types of differential equations.

• Ordinary Differential Equations
• Partial Differential Equations

## Ordinary Differential Equations

### Definition

An equation containing the derivative of only one dependent variable (unknown function) with respect to only one independent variable is called an ordinary differential equation. It is denoted by (ODE).

### Example

dy/dx +5y = e^x

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d^2y/dx^2 +6 = 0

dx/dt + dy/dt + 9 = 0

These are all examples of ordinary differential equations.

## Partial Differential Equations

An equation containing the derivative of one or more dependent variables (unknown functions) with respect to two or more independent variables is called a partial differential equation. It is denoted by (PDE).

### Examples

d^2u/dx^2 = 0

d^2u/dx^2 = d^2u/dt^2

## Classification by Order

The order of a differential equation either O.D.E or P.D.E is the order of the highest derivative in the equation.

### Example

d^2y/dx^2+5(dy/dx)=e^x

## Classification by Linearity or Non-Linearity

A differential equation may be linear or non-linear.

### Linearity of a Differential Equation

A differential equation is said to be linear if it satisfies the following two properties,

1) The dependent variable y and all its derivatives are of the first degree, that is the power of each term involving y is 1.

2) The co-officiant of the dependent variable depends at most on the independent variable x.

### Non-Linearity of a Differential Equation

If a differential equation is not linear, then it is said to be a non-linear differential equation.

### Examples

Linear Differential Equations

d^2y/dx^2-2y = 0

X^3(d^3y/dx^3)-x (dy/dx) = 0

Non-Linear Differential Equations

(1-y) dy/dx + 2y = e^x

## Applications of Ordinary Differential Equations

Differential equations are used in everyday life to compute the flow of electricity, the motion of an object back and forth like a pendulum, and to illustrate the principles of thermodynamics. Additionally, in medical terminology, they are employed to graphically monitor the progression of diseases.

Many distinct natural phenomena, including sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, and quantum physics, can be explained by differential equations. Differential equations, for instance, can be used to simulate the motion of particles in a fluid or the trajectory of a projectile in physics.

In order to study the behavior of the systems in problems, engineers must be able to represent physical issues using mathematical equations and then solve those equations. In order to measure temperature variation to calculate the rate of travel, such as miles per hour, kilometers per hour, etc. Numerous equations in physics are derived using derivatives.

Finding the range of earthquake magnitudes is a popular task in seismology research. First-order differential equations are mostly used to study growth and decay, Newton's cooling rule, and dilution issues. Different partial derivatives of an unidentified multivariable function are related by partial differential equations.

The mathematical formulation of physical and other issues requiring functions of several variables, such as the transmission of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc., is facilitated by the use of differential equations.

Differential equations are used in the medical field as well. For instance, differential equations can be used to estimate a drug's course over time. In differential equation applications, the functions represent physical values, and the derivatives, as we know, reflect the rates at which these qualities change.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

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