 A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form 
where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable .Among ordinary differential equations, linear differential equations play a prominent role for several reasons.Most elementary and special functions that are encountered in physics and applied mathematics are solutions of linear differential equations (see Holonomic function).