Advanced Calculus fi..

648 Advanced Calculus, Fifth EditionHence there is a recursion formula for the coefficients:Thusc2=--,co co coCq= - c6=--2 2.4 2.4.6'""It appears that co and c, are arbitrary constants; they are, in fact, simply the initialvalues of y and y' at x = 0. The solutions found can now be written in the form:Here an application of the ratio test shows that the series converge for all x. Furthermore,y satisfies the differential equation for all x. Since the functionsare clearly linearly independent, the functions- ractually give the general solution of the differential equation.This method can also be used for nonlinear equations, but the algebraic processesusually become highly involved, and the chances of obtaining the general term ofthe series or an expression for the general solution are very small.For many applications it is important to have a series solution of a differentialequation y' = F(x, y), even when the function F(x, y) is not analytic in a neighborhoodof the initial point considered. The most common case is that in which Fis a rational function whose denominator is 0 at the initial point. The initial point isthen a singular point of the differential equation, and no general statement can bemade about solutions. However, it is possible in many cases to obtain the solutionsthrough and near the singular point in the form of series of appropriate types. Forexample, the serieswhere m is not necessarily positive or an integer, can be used in certain cases. Inother cases the solution can be expressed as a series- ,where m and p are quite general.Similar remarks apply to equations of higher order and to systems of equations.Series of type (9.51) are of special importance for linear equations. For example,