Advanced Calculus fi..

Ordinary DifferentialEquations9.1 DIFFERENTIAL EQUATIONSAn ordinary differential equation of order n is an equation of form* '+expressing a relation between x, an unspecified function y(x), and its derivativesy', y", . . . through the nth order. For example,are ordinary differential equations of orders 2 and 3, respectively.In order for the differential equation (9.1) to have significance, it is necessarythat the function F be defined in some domain of the space of the variables on whichit depends. In this chapter we shall consider only equations that can be solved forthe highest derivatives and written in the form:y(n) = F(x, y, y', . . . , y'n-").Equation (9.2) is at once reducible to this form. Equation (9.3) is a quadratic equationin y"'; by solving this quadratic equation we obtain two different equations for y"',that is, two equations of form (9.4). We say that (9.3) is a differential equation ofdegree 2, whereas (9.2) is of degree 1.Systems of differential equations will also be considered (Section 9.5).4(9.4)625