An nth-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. As examples, (4) is linear while (2), (3), and (5) are nonlinear. Equation (1) is linear when is a linear function of u; otherwise, it is nonlinear. 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. An important property of linear equations is that of superposition: To illustrate the superposition principle, consider the following IVP:
The data for this problem are . 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 nth order differential equations. In practice, superposition permits us to decompose a problem with complicated data into simpler parts, to solve each problem separately, and then to combine these solutions to find the solution to the original problem.