By S. L. Sobolev and A. J. Lohwater (Auth.)
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Touch, and there are no such points by hypothesis. Dividing the equation by Ai throughout, the principal term becomes 5 2 w/3| 2 . 14) does not define a single curve but only a family of curves?? = const, and so a certain arbitrariness remains for the choice of the function ξ(χ, y). In the case when our second-order partial differential equation is of elliptic or hyperbolic type, it is easy to show that the coefficient Bx vanishes. 02 C A N O N I C A L FORMS: T W O I N D E P E N D E N T V A R I A B L E S 41 The complete reduction to canonical form of an equation of elliptic type is a complicated problem, and we shall not stop to deal with it.
2. Canonical Form of Equations in Two Independent Variables As we have seen, an equation with variable coefficients can always be brought into canonical form at an isolated point in space. If the number of independent variables is greater than two, then only in exceptional circum stances can the equation be brought into canonical form throughout the whole space in which the equation is given. However, an equation in two independent variables can always be brought into canonical form throughout the whole region of variation of the independent variables.
6) and also [ 4 = o = 0 and [u]x=l = 0. Obviously the functions