Nonlinear Differential Equations

1. Introduction. A few nonlinear differential equations have known exact solutions, but many which are important in applications do not. Sometimes these equations may be linearized by an expansion process in which nonlinear terms are discarded. When nonlinear terms make vital contributions to the solution this cannot be done, but sometimes it is enough to retain a few "small" ones. Then a perturbation theory may be used to obtain the solution. A differential equation may sometimes be approximated by an equation with "small" nonlinearities in more than one way, giving rise to different solutions valid over different ranges of its parameters. There are two types of small nonlinearity problems. In the first type the nonlinearities occur in the most highly differentiated terms. These are very important in several physical theories. Carrier refers to them as "boundary layer problems" [l; 2] in recognition of the application in which they had their first important development. They include many important nonlinear partial differential equations problems, as well as some ordinary nonlinear differential equations in which such phenomena as relaxation oscillations occur. Boundary layer problems are usually closely tied in with applications. Their theories have not yet received very general or exhaustive development , and much art and ingenuity has been called for in the work that has been done ([l]-[3]). The second type of nonlinearity problem is that in which non-linearities do not occur in the most highly differentiated terms. In this case the theory has been developed farther, and something more nearly resembling a general method of attack is possible. Actually several such methods have been developed. Each has its own special merits and limitations. I will discuss one such method. This particular method has the advantage of wide scope and practicality of application , but is limited to a class of differential equations which is associated with nonconservative physical systems. This method offers nothing new in the case of ordinary nonlinear differential equations of the second order, but has a practical advantage in the case of systems of equations (or, what comes to the same