Digression on Generalizations of the Geometric Method of Solution. — Solutions of Equation (1.1) for the Domain x > 0 with a Boundary Condition at x = 0

Abstract

In the preceding section it has been found that for small values of v the solution of Equation (1.1) can be obtained by means of a geometric construction. This construction makes use of two curves, one of which glides over the other one. The latter curve, the ‘s-curve’, is dependent upon the initial conditions connected with Equation (1.1), and contains a representation of the prescribed initial course u ₀(x). When a solution is sought for another initial course, this ‘s-curve’ has to be changed. The other curve, the ‘S-curve’, which glides over the s-curve, is not dependent upon the initial conditions. It can be considered as a kind of ‘sensing mechanism’, or ‘resolvent’, determined by the nature of the differential equation. The question can be raised whether there are other equations, which give rise to a similar procedure for obtaining a solution.