Range of the Chapter. The general theory of linear ordinary differential equations with variable coefficients covers an immense field which is quite beyond the scope of this book. In the present chapter our principal purpose will be to indicate how matrices can be applied usefully in the approximate solution of such differential equations.

Amongst the various special methods discussed, those described in § 7·9 and exemplified in § 7·10 are particularly powerful: one important field of application is to problems in mechanics which involve the determination of natural frequencies. The method of mean coefficients, described in § 7·11, is somewhat laborious, but it leads to good approximations even when the true solution is highly oscillatory. Examples of the use of this method are given in §§ 7·12–7·15.

Existence Theorems and Singularities. Linear differential equations with variable coefficients are rarely soluble by exact or elementary methods, and it is usually necessary to resort to numerical approximations. In a one-point boundary problem, for example, where the values of the dependent variable (or variables) and of the derivatives up to a certain order are specified at some datum point, say t = t0, the normal procedure is to try a development of the solution in the form of a series of ascending powers of t − t0. It is obviously of great assistance to know beforehand whether such a form of solution is justified, and, if so, the range of convergence. This information is supplied by “existence theorems”, which specifically concern the conditions to be satisfied in order that solutions of differential equations may exist, and the ranges of validity of such, solutions.