It is known [Herman Weyl, 1910] that every linear second-order differential expression L (with real coefficients) is such that Ly = λy (im λ ≠ 0) has at least one solution belonging to the class L2 = L2[0, ∞) of functions, the squares of whose moduli are Lebesgue-integrable on [0, ∞). This celebrated result was later proved by E. C. Titchmarsh (1940–1944), using sophisticated analysis of bilinear transformation. The aim of the present note is to prove the same result once again, but using only elementary analysis and school geometry. The power of this method will be appreciated further when one realises the amount of simplifications that can be acheived by this expressions. This part of the note course will be taken up in a subsequent paper.

The study of plasma instabilities has led to the question whether a certain third order linear differential equation involving a parameter p has solutions which vanish as x → ± ∞. Assuming existence, it is first easily shown that Rep must be positive and then, after a Fourier transform has changed the equation to one of second order, standard comparison equation techniques are used to obtain a contradiction, valid for large enough p.

We shall be concerned with two boundary value problems for the Falkner-Skan Equation

when –β is a small positive number. Our interest is in solutions of (1) which exhibit “reversed flow”; that is, solutions f such that f′(x) < 0 for small positive values of x. The boundary conditions which we wish to consider are

and

In this note we report on some numerical results regarding reverse flow solutions of the Falkner-Skan equation

on the half line 0 < t < ∞, which satisfy the boundary conditions

and

The study of similarity solutions of Prandtl's equations for the steady two dimensional flow of an incompressible fluid past a rigid wall leads to the equation

where the primes denote differentiation with respect to the independent variable t, and λ is a parameter. It was first obtained in 1930 by Falkner and Skan [3]. For its derivation we refer to Schlichting [6] here we merely note that the function f′(t) represents, after suitable normalization, the velocity parallel to the wall.