what is the difference between differentiate and derivate?

Differentiation is a process. Derivative is the result obtained from differentiation.

y=2x3

dydx=6x2

is the derivative obtained by differentiation.

differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables relating:

• The values of the function itself, and
• Its derivatives of various orders.

Order

The order of a differential equation is defined as being that of the highest order derivative that is present in the equation.

ODE and PDE

Ordinary Differential Equation

An ordinary differential equation (abbreviated O.D.E. or ODE) is one which has only one independent variable.

All the derivatives occurring in it are therefore ordinary. 

Partial Differential Equation

A partial differential equation (abbreviated P.D.E. or PDE) is one which has more than one independent variable.

The derivatives occurring in it are therefore partial.

linear and non linear

Linear

A linear differential equation is one where any dependent variables and their derivatives appear to the first power.
Neither are products of dependent variables allowed.

Non-Linear

A non-linear differential equation is one which is not linear.