Advanced Calculus fi..

626 Advanced Calculus, Fifth EditionThe term ordinary is used here to emphasize that no partial derivatives appearsince there is just one independent variable. An equation such aswould be called a partial diflerential equation. Chapter 10 is devoted to such equations.The applications of ordinary differential equations to physical problems are numerous.The equations of dynamics are relations between coordinates, velocities,accelerations, and time and hence give differential equations of second order or systemsof higher order. Electric circuits obey laws described by differential equationsrelating currents and their time derivatives. Servomechanisms, or control systems,are combinations of mechanical and electrical (and perhaps other) components andcan be described by differential equations. Problems involving continuous mediafluiddynamics, elasticity, heat conduction, and the like-lead to partial differentialequations.By a solution or particular solution of (9.1) is meant a function y = f(x),a < x < b, having derivatives up to the nth order throughout the interval and suchthat (9.1) becomes an identity when y and its derivatives are replaced by f (x) andits derivatives. Thus y = ex is a particular solution of (9.2), and y = x is a particularsolution of (9.3). For most of the differential equations to be considered here it willbe found that all particular solutions can be included in one formula:where el, . . . , en are "arbitrary" constants. Thus for each special assignment ofvalues to the c's, (9.6) gives a solution of (9. I), and all solutions can be so obtained.(The range of the c's and of x may have to be restricted in some cases to avoidimaginary expressions or other degeneracies.) For example, all solutions of (9.2) aregiven by the formulay = cle-x + c2e-2X + ex; (9.7)the solution y = ex is obtained when c, = 0, c2 = 0. When a formula such as (9.6)is obtained, providing all solutions, it is called the general solution of (9.1). For theequations considered here it will be found that the number of arbitrary constantsequals the order n.The presence of arbitrary constants should not be surprising, for they occur inthe simplest differential equation:All solutions of (9.8) are obtained by integration: