Advanced Calculus fi..

Chapter 9 Ordinary Differential Equations 647EXAMPLEy' = x2 - y2; y = 1 for x = 0. HereHence at x = 0, one has, for the solution sought, -. ,Thus the solution is given byIt is essentially hopeless to attempt to find the general term here; this is a characteristicdefect of the method when it is applied to a nonlinear differential equation. However,the general theorem referred to earlier gives assurance that the series does convergefor x in some interval -a < x < a and is a solution. Furthermore, it is possible toobtain estimates for the interval of convergence and for the remainder after n terms.Hence the series can be used for carefully controlled numerical work.The method described applies equally well to higher-order differential equationsand to systems of equations. Thus for the equationwith initial conditions y = 1 and y' = 2 for x = 1, one hasso thaty"' = 2yy' + I, y" = 2y'2 + 2yy", . . . ,For linear differential equations with variable coefficients, another way of obtainingthe series is available and will often make it easy to find an expression forthe general term of the series. This is a method of "undetermined coefficients." Forexample, let the equation bey"+xyf+y=o,and let the solution be sought in the form of a series about x = 0:Upon substituting this series in the equation and collecting terms according to powersof x, one obtains the equation:By the corollary to Theorem 40 of Section 6.16, each coefficient must equal 0. Henceone obtains the equations: