Mathematical Methods for Physics and Engineering - Matematica.NET

15Higher-order ordinary differentialequationsFollowing on from the discussion of first-order ordinary differential equations(ODEs) given in the previous chapter, we now examine equations of second andhigher order. Since a brief outline of the general properties of ODEs and theirsolutions was given at the beginning of the previous chapter, we will not repeatit here. Instead, we will begin with a discussion of various types of higher-orderequation. This chapter is divided into three main parts. We first discuss linearequations with constant coefficients and then investigate linear equations withvariable coefficients. Finally, we discuss a few methods that may be of use insolving general linear or non-linear ODEs. Let us start by considering somegeneral points relating to all linear ODEs.Linear equations are of paramount importance in the description of physicalprocesses. Moreover, it is an empirical fact that, when put into mathematicalform, many natural processes appear as higher-order linear ODEs, most oftenas second-order equations. Although we could restrict our attention to thesesecond-order equations, the generalisation to nth-order equations requires littleextra work, and so we will consider this more general case.A linear ODE of general order n has the forma n (x) dn ydx n + a n−1(x) dn−1 ydx n−1 + ···+ a 1(x) dydx + a 0(x)y = f(x). (15.1)If f(x) = 0 then the equation is called homogeneous; otherwise it is inhomogeneous.The first-order linear equation studied in subsection 14.2.4 is a special case of(15.1). As discussed at the beginning of the previous chapter, the general solutionto (15.1) will contain n arbitrary constants, which may be determined if n boundaryconditions are also provided.In order to solve any equation of the form (15.1), we must first find thegeneral solution of the complementary equation, i.e. the equation formed by setting490