Theory of Partial Differential Equations

Abstract

In what follows, any letter not otherwise explained denotes a function of certain variables (x, y, p, q), or (x, y, z, p, q, r), &c., as will be stated in each particular case.An equation a = const.denotes that the function a of the variables is, in fact, a constant (viz.by such equation we establish a relation between the variables): and when this is so, we use the same letter a to denote the constant value of the function in question; I find this a very convenient notation.Thus if the variables are x, y, z, p, q, r and if p, q, r are the differential coefficients in regard to x, y, z respectively of a function V of x, y, z, then H (as a letter not otherwise explained) denotes a function of x, y, z, p, q, r and considering it as a given function, H = const.will be a partial differential equation containing the constant H.For instance, if H denote the function pqr -xyz, H = const.is the partial differential equation, pqrxyz = H (a given constant).The integration of the partial differential equation, H = const., depends upon that of the linear partial differential equation (H, Θ) = 0, where as usual (Η, Θ) signifies It can be effected if we know two conjugate solutions a, b of the equation (H, Θ) = 0, viz.a, b as solutions are such that (H, α) = 0, (H, b) = 0, and (as conjugate solutions) are also such that (a, b) = 0; in this case if from the equations H = const., a = const., b = const.

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