Mathematical Methods for Physics and Engineering - Matematica.NET

14First-order ordinary differentialequationsDifferential equations are the group of equations that contain derivatives. Chapters14–21 discuss a variety of differential equations, starting in this chapter andthe next with those ordinary differential equations (ODEs) that have closed-formsolutions. As its name suggests, an ODE contains only ordinary derivatives (nopartial derivatives) and describes the relationship between these derivatives ofthe dependent variable, usually called y, with respect to the independent variable,usually called x. The solution to such an ODE is therefore a function of x andis written y(x). For an ODE to have a closed-form solution, it must be possibleto express y(x) in terms of the standard elementary functions such as exp x, lnx,sin x etc. The solutions of some differential equations cannot, however, be writtenin closed form, but only as an infinite series; these are discussed in chapter 16.Ordinary differential equations may be separated conveniently into differentcategories according to their general characteristics. The primary groupingadopted here is by the order of the equation. The order of an ODE is simply theorder of the highest derivative it contains. Thus equations containing dy/dx, butno higher derivatives, are called first order, those containing d 2 y/dx 2 are calledsecond order and so on. In this chapter we consider first-order equations, and inthe next, second- and higher-order equations.Ordinary differential equations may be classified further according to degree.The degree of an ODE is the power to which the highest-order derivative israised, after the equation has been rationalised to contain only integer powers ofderivatives. Hence the ODEd 3 ( )y dy 3/2dx 3 + x + x 2 y =0,dxis of third order and second degree, since after rationalisation it contains the term(d 3 y/dx 3 ) 2 .The general solution to an ODE is the most general function y(x) that satisfiesthe equation; it will contain constants of integration which may be determined by468