# What is differential equation with examples ? Differential equation and it's types

## Differential equation

An equestion that is consist of derivative of one or more dependent or unknown variables wrt one or more independent or known valuables is called differential equations.

Examples;

1 - dy/dX + 10y = e²
2- d²y/dx² + dy/dX + 8y =0
3 - d⁴y/dx⁴ - d³y/dx³ + d²y/dx² = 0

# Types of differential equation

There two basics types of differential equation;

ODE

Ordinary differential equation
PDE

Partial differential equation

## Ordinary differential equation

If a differential equation consist of derivative ( ordinary derivative ) of one or more unknown (dependent variables) with respect to a single known (independent variable) then it is called ordinary differential equation.

Examples;

1 – dy/dx – Y = X + 7Y
2 - d⁵y/dx⁵ + X = Y – 4X
3 - d²y/dx² + dy/dx -1 = 0

## Partial differential equation

If there is a differential equation that is consist of derivative of one or more dependent variables (unknown) wrt two or more independent variables (known) then it is known as partial differential equation.

Examples;

1 - &²u/&x² + &²u/&y² = 0
2 - &²u/&x² = &²u/&t²

As it clear from name partial differential equation that it is consist of partial derivatives.
Here the sign & represent partial sign.

# Classification of differential equation by order

## Order of differential equation

The order of highest derivative in ordinary differential equation or partial differential equation is called order of differential equation.(ordinary or partial)

Examples;

1 - d²y/dx² - (dy/dx)⁴ + y = 2x

Here order of ODE is two because it consist of second order dy²/dx² but not (dy/DX)⁴.

2 - &³t/&x³ + &²t/&y² + (&t/&z)⁴ = 0

Here order of PDE is three because it highest order is 3 but not highest power that is 4.

# Notations for differential equations

Differential equation can be written by using Leibniz notation such as dy/dX, d²y/dx², .... Or it can be written by using prime notations such as y’,y”,y”’,... But there is a limit to use prime because we can represent first, second and third order differential equations by prime notations and higher derivative differential equations can be written as y⁴ , y⁵ ,...