The order of a differential equation is the order of the highest derivative appearing in the equation. In this case, is also referred to as a general solution because all solutions to the differential equation can be represented in this form for appropriate choices of the constants and . The function is a particular solution because it contains no arbitrary constants. With a differential equation, we can associate initial conditions or boundary conditions , auxiliary conditions on the unknown function and its derivatives. An th-order differential equation is said to be linear if it can be written in the form A nonlinear differential equation is simply one that is not linear.
Differential equations arising from first principle models are generally nonlinear.
Nonlinear equations do not usually yield to analytical approaches and computational methods are called for. Linear equations constitute a highly important class of differential equations in physics and engineering and are used in idealized models of such phenomena as mechanical vibrations, electrical circuits, planetary motions, etc. If is a solution with data , and is a solution with data , then the principle states that is a solution for the data . This idea extends readily to th order differential equations. If the diameter of the tube approaches zero as becomes large, then is said to be asymptotically stable .