Note on Two Methods of Solving Ordinary Linear Differential Equations

Two assumptions are formulated, based on recent results concerning two methods of approximating to the solutions of ordinary linear differential equations. They are shown to be false by means of counter examples. 1. An important class of methods for finding global solutions to ordinary linear differential equations involves assuming a trial solution containing free parameters, and determining these by some strategy. Attention has recently focussed on two methods of this kind: (a) methods equating coefficients of the independent variable, (b) collocation methods, where the residual is made equal to zero at certain values of the independent variable. In this note we consider two assumptions based on recent results regarding these methods, and show them to be false by means of counter examples. 2. We consider first the most common method of type (a), viz. the Lanczos r-method (Lanczos 1957). Here we obtain polynomial approximations to the solution of the linear differential equation