Essentials of Mathematical Methods in Science and Engineering, 2nd Edition

CHAPTER 11ORDINARY DIFFERENTIAL EQUATIONS

Differential equations are proven to be very useful in describing the majority of the physical processes in nature. They are composed of the derivatives of an unknown function, which could depend on several variables. This means that the corresponding differential equation contains partial derivatives of the unknown function. Such equations are called partial differential equations. They are in general quite difficult to solve and deserve separate treatment. On the other hand, in a lot of the physically interesting cases, symmetries of the system allow us to eliminate some of the variables, thus reducing a partial differential equation into an ordinary differential equation. In this chapter, we discuss ordinary differential equations in detail and concentrate on the methods of finding analytic solutions in terms of known functions like polynomials, exponentials, trigonometric functions, etc. We start with a discussion of the first‐order differential equations. Since the majority of the differential equations encountered in applications are second‐order, we give an extensive treatment of the second‐order differential equations and introduce techniques of finding their solutions. The general solution of a differential equation always contains some arbitrary parameters called the integration constants. To facilitate the evaluation of these constants, differential equations have to be supplemented with additional information called the boundary ...