Mathematical Methods for Physics and Engineering - Matematica.NET

14.1 GENERAL FORM OF SOLUTIONthe application of some suitable boundary conditions. For example, we may betold that for a certain first-order differential equation, the solution y(x) isequaltozero when the parameter x is equal to unity; this allows us to determine the valueof the constant of integration. The general solutions to nth-order ODEs, whichare considered in detail in the next chapter, will contain n (essential) arbitraryconstants of integration and therefore we will need n boundary conditions if theseconstants are to be determined (see section 14.1). When the boundary conditionshave been applied, and the constants found, we are left with a particular solutionto the ODE, which obeys the given boundary conditions. Some ODEs of degreegreater than unity also possess singular solutions, which are solutions that containno arbitrary constants and cannot be found from the general solution; singularsolutions are discussed in more detail in section 14.3. When any solution to anODE has been found, it is always possible to check its validity by substitutioninto the original equation and verification that any given boundary conditionsare met.In this chapter, firstly we discuss various types of first-degree ODE and then goon to examine those higher-degree equations that can be solved in closed form.At the outset, however, we discuss the general form of the solutions of ODEs;this discussion is relevant to both first- and higher-order ODEs.14.1 General form of solutionIt is helpful when considering the general form of the solution of an ODE toconsider the inverse process, namely that of obtaining an ODE from a givengroup of functions, each one of which is a solution of the ODE. Suppose themembers of the group can be written asy = f(x, a 1 ,a 2 ,...,a n ), (14.1)each member being specified by a different set of values of the parameters a i .Forexample, consider the group of functionsy = a 1 sin x + a 2 cos x; (14.2)here n =2.Since an ODE is required for which any of the group is a solution, it clearlymust not contain any of the a i . As there are n of the a i in expression (14.1), wemust obtain n + 1 equations involving them in order that, by elimination, we canobtain one final equation without them.Initially we have only (14.1), but if this is differentiated n times, a total of n +1equations is obtained from which (in principle) all the a i can be eliminated, togive one ODE satisfied by all the group. As a result of the n differentiations,d n y/dx n will be present in one of the n + 1 equations and hence in the finalequation, which will therefore be of nth order.469