PARTIAL DIFFERENTIAL EQUATIONS
A differential equation involving partial derivatives
of a dependent variable(one or more) with
more than one independent variable is called a partial differential
equation, hereafter denoted as PDE.
Consider the following equations:

(6.1.1) 

(6.1.2) 

(6.1.3) 

(6.1.4) 

(6.1.5) 

(6.1.6) 
Order of a PDE: The order of the highest
derivative term in the equation is called the order of the PDE. Thus
equations (6.1.1 to 6.1.6) are all of second order.
Linear PDE: If the dependent variable
and all its partial derivatives occure linearly in any PDE then such
an equation is called linear PDE otherwise a nonlinear PDE. In the
above example equations 6.1.1, 6.1.2, 6.1.3 & 6.1.4 are linear whereas
6.1.5 & 6.1.6 are nonlinear.
Quasilinear PDE: A PDE is called as
a quasilinear if all the terms with highest order derivatives of dependent
variables occur linearly, that is the coefficients of such terms are
functions of only lower order derivatives of the dependent variables.
However, terms with lower order derivatives can occur in any manner. Equation 6.1.5
in the above list is a Quasilinear equation.
Homogeneous PDE: If all the terms of
a PDE contains the dependent variable or its partial derivatives then
such a PDE is called nonhomogeneous partial differential equation or homogeneous otherwise.
In the above six examples eqn 6.1.6 is nonhomogeneous where as the first
five equations are homogeneous.
Notation: It is also a common practise
to use subscript notation in writing partial differential equations.
For example the Laplace Equation in three dimensional space

(6.1.7) can be wriiten as 

(6.1.8) 
