Periodic differential equations

Arscott F.M.

Of recent years, the two subjects of special functions and ordinary linear differential equations have (in this country at least) lain somewhat in the penumbra of a partial eclipse. Many mathematicians, not without reason, have come to regard the former as little more than a haphazard collection of ugly and unmemorable formulae, while attention in the latter has been directed mostly to existence-theorems and similar results for equations of general type. Only rarely does one find mention,at post-graduate level, of any problems in connection with the process of actually solving such equations. The electronic computer may perhaps be partly to blame for this, since the impression prevails in many quarters that almost any differential equation problem can be merely ”put on the machine”, so that finding an analytic solutionis largely a waste of time. This, however, is only a small part of the truth, for at the higher levels there are generally so many parameters or boundary conditions involved that numerical solutions, even if practicable, give no real idea of the properties of the equation. Moreover, any analyst of sensibility will feel that to fall back on numerical techniques savours somewhat of breaking a door with a hammer when one could, with a little trouble, find the key.