# Differential Equations

A **Differential Equation** is a mathematical equation involving derivatives of an unknown function or functions, which may be dependent on one or more variables. These equations are fundamental in the fields of engineering, physics, biology, economics, and many other disciplines as they often describe the dynamics of various systems. The goal is often to find the unknown function that satisfies the given equation. The study of differential equations encompasses methods to solve these equations and to understand the properties of the solutions.

## Types of Differential Equations

### Ordinary Differential Equations

Ordinary Differential Equations (ODEs) involve derivatives with respect to only one independent variable. They can be classified as:

- First-Order: Involving only the first derivative
- Second-Order: Involving up to the second derivative
- nth-Order: Involving derivatives up to the nth order

Further, they can be:

- Linear or Non-linear: Depending on whether the equation can be written in a linear form.
- Homogeneous or Non-homogeneous: Depending on the presence of a term not involving the unknown function or its derivatives.

### Partial Differential Equations

Partial Differential Equations (PDEs) involve partial derivatives with respect to more than one independent variable. They appear often in fields like fluid dynamics, electromagnetism, and quantum mechanics. PDEs can also be linear or non-linear and can be classified based on the highest order of the derivative involved.

## Methods of Solution

### Analytical Methods

Analytical methods aim to find an exact solution to the differential equation. These methods include:

- Separation of Variables
- Method of Characteristics
- Laplace Transforms
- Series Solutions
- Green's Functions

### Numerical Methods

When an analytical solution is not possible or is overly complex, numerical methods can be employed to approximate the solution. Methods include:

- Euler's Method
- Runge-Kutta Methods
- Finite Difference Method
- Finite Element Method

## Applications

Differential equations find applications in a wide array of disciplines:

- Physics: Describe motion, electricity, heat, and quantum mechanics
- Engineering: Used in control theory, signal processing, and mechanical simulations
- Biology: Model populations, spread of diseases, and gene expression
- Economics: Used to model economic systems
- Computer Graphics: For simulations in graphics

## History

The foundations of differential equations were laid in the 17th century with the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz. Over the centuries, many mathematicians including Euler, Laplace, and Fourier have contributed to the field.