Definitions
In what follows, let y be a dependent variable and x an independent variable, so that y = y(x) is an unknown function in x. The notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. In this context the Leibniz's notation (dy/dx,d2y/dx2,...dny/dxn) is useful for differentials and when integration is to be done, while Newton's and Lagrange's notation (y′,y′′, ... y(n)) is useful for representing derivatives of any order compactly.
Read more about this topic: Ordinary Differential Equation
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