Type of the Differential Equations

Odinary Differential Equations

An Ordinary Differential Equation is a differential equation that depends on only one independent varialble.
For example
is an Odinary Differential Equation because y(the independent variable) depends only on t(the independent variable)

Partial Differential Equations

A Partial Differential Equation is differential equation in which the dependent varialble depends on two or more independent variables.
For example
The Laplace's equation is a Partial Differential Equation because f depends on two independent variables x and y.

Order of a Differential Equation

The order of a differential is the order of the highest derivative entering the equation. 
For example
The equation is called a second-order differential equation because it involves second derivatives.

Linear Differential Equation

A first-order differential equation is linear if it can be written in the form where g(t) and r(t) are arbitary functions of t.
For example
is a first-order linear differential equation where and

Nonlinear Differential Equation

It is a differential equation whose right hand side is not a linear function of the dependent variable.
For example

Homogeneous Differential Equation

A linear first-order differential equation is homogeneous if its right hand side is zero , that is

For example
, where k is a constant, is homogeneous.

Nonhomogeneous Differential Equation

A linear first-order differential equation is nonhomogeneous if its right-hand side is non-zero that is .

For example
is nonhomogeneous.